Proof. For any integer B, the statement ‘the complexity of the connected component of an algebraic group $\mathbf {L}$ is at most B’ is a first-order statement on the coefficients of the polynomials defining $\mathbf {L}\subset (GL_n)_{F}$ .

The result follows now by ultraproduct argument.

Proof.

1. 1. Let ${\mathcal S}'$ be the domain of definition of $\pi $ and let ${\mathcal G}_{{\mathcal S}'}:={\mathcal G}\times _{\mathcal S} {\mathcal S}'$ . Define

   $$ \begin{align*}{\mathcal K}:={\mathcal H} \times_{{\mathcal G}_{{\mathcal S}'}} \cdots \times_{{\mathcal G}_{{\mathcal S}'}}{\mathcal H}\end{align*} $$
   and
   $$ \begin{align*}{\mathcal S}":={\mathcal S}' \times_{{\mathcal S}} \cdots \times_{{\mathcal S}'} {\mathcal S}'\end{align*} $$
   to be the d-fold fibered products. The natural maps $\mathcal {S}" \rightarrow \mathcal {S}$ and ${\mathcal K}\to {\mathcal S}"$ give a subfamily as required.

2. 2. After passing to a stratification of $\mathcal {S}$ , we can assume that $\mathcal {G} \subset (\operatorname {\mathrm {GL}}_n)_{\mathcal {S}}$ is closed. Fix a closed embedding $(\operatorname {\mathrm {GL}}_n)_{\mathcal {S}} \hookrightarrow \mathbb {A}_{\mathcal {S}}^N$ .

   Since $\mathcal {G} \rightarrow \mathcal {S}$ is of finite type, there is a bound D on the complexity of all subgroups $\mathcal {G}_{s}$ , where s ranges over all geometric points of $\mathcal {S}$ . By Lemma 4.0.6, there is a bound E on the complexity of all subgroups $(\mathcal {G}_{s})^\circ $ , where s ranges over all geometric points of $\mathcal {S}$ . Since $\mathcal {G} \rightarrow \mathcal {S}$ is of finite type, there is a constant C such that, for any geometric point s of $\mathcal {S}$ , the group $\mathcal {G}_{s}$ has at most C connected components. We get that there is a constant M such that for any geometric point $s\in {\mathcal S}$ , any open subgroup of $\mathcal {G}_s$ has complexity at most M.

   There is a morphism $\mathcal {T} \rightarrow \mathcal {S}$ and a family of subgroups $\mathcal {P} \subset \mathcal {G}_{\mathcal {T}} \rightarrow \mathcal {T}$ such that the following holds: for every prime power q, every $s\in \mathcal {S}(\mathbb {F}_q)$ and every algebraic subgroup $\mathcal {P} \subset \mathcal {G}_s$ which is defined over $\mathbb {F}_q$ and has complexity at most M, we have that $\mathcal {P}=\mathcal {P}_t$ , for some $t\in \mathcal {T}(\mathbb {F}_q)$ . By the arguments above, this family satisfies the requirements.

Proof of Corollary 4.0.4.

It is enough to construct $\mathcal {K}$ and prove that the claim holds for all but finitely many primes. Let $\pi :\mathcal {S}'\rightarrow \mathcal {S}$ and $\mathcal {H} \rightarrow \mathcal {S}'$ be as in Theorem 4.0.3 and let $d={\operatorname {dim}}_{\mathcal S}\, \mathcal {G}$ .

By Lemma 4.0.7, we have a family $\mathcal {K}\Subset _{\tau } \mathcal {G}$ , with $\tau :{\mathcal T}\to {\mathcal S}$ , such that, for any geometric point s of $\mathcal {S}$ and any d geometric points $s_1,\ldots ,s_d$ of $\pi ^{-1} (s)$ , any open subgroup of $\mathcal {H}_{s_1} \cap \cdots \cap \mathcal {H}_{s_d}$ is of the form $\mathcal {K}_t$ , where t is a point over s (i.e., $\tau (t)=s$ ). We show that such a family $\mathcal {K}$ satisfies the conclusion of the Corollary.

Let q be a prime power, $s\in \mathcal {S}(\mathbb {F}_q)$ , and $\Gamma \subset \mathcal {G}_s(\mathbb {F}_q)$ that $\mathbb {F}_q$ -evades $\mathcal {K}$ . Let $F: \pi ^{-1}(s)\to \pi ^{-1}(s)$ be the geometric Frobenius. We first show that $\Gamma \ \overline {\mathbb {F}_q}$ -evades $\mathcal {H}$ . Assuming the contrary, there is $s' \in \pi ^{-1} (s)(\overline {\mathbb {F}_q})$ such that $\Gamma \subset \mathcal {H}_{s'}(\overline {\mathbb {F}_q})$ . For every finite subset $I \subset \mathbb {Z}$ , denote $\mathbf {H}_I=\bigcap _{i \in I} \mathcal {H}_{F^is'}$ . There are $i_1,\ldots ,i_d$ such that ${\operatorname {dim}}\, \mathbf {H}_{\{i_1,\ldots ,i_d\}}=\min \left\{ {\operatorname {dim}}\, \mathbf {H}_I \mid I \subset \mathbb {Z} \text { finite}\right\}$ . The group $\mathbf {H}_{\{i_1,\ldots ,i_d\}}^\circ $ is invariant under the Frobenius, so it is defined over $\mathbb {F}_q$ . It follows that the group $\Gamma \mathbf {H}^{\circ }_{\{i_1,\ldots ,i_d\}}$ is also defined over $\mathbb {F}_q$ . By the assumption on $\mathcal {K}$ , we have $\Gamma \mathbf {H}_{\{i_1,\ldots ,i_d\}}^\circ = \mathcal {K}_t$ , for some $t\in \mathcal {T}(\mathbb {F}_q)$ . This contradicts the assumption that $\Gamma \ \mathbb {F}_q$ -evades $\mathcal {K}$ .

The result now follows from Theorem 4.0.3.

Proof. Let $\widetilde {\mathbf {H}}$ be the universal cover of $\mathbf {H}$ . By Theorem 3.1.1, the universal central extension (or universal cover; cf. [Reference MooreMoo68, §1]) of $\mathbf {H}(\mathbb {F}_p)'$ is $\widetilde {\mathbf {H}}(\mathbb {F}_p)$ . Denote the kernel of $\widetilde {\mathbf {H}}(\mathbb {F}_p) \rightarrow \mathbf {H}(\mathbb {F}_p)' $ by A and note that

$$ \begin{align*}A \subset Z(\widetilde{\mathbf{H}}(\mathbb{F}_p)) \overset{{3.1.1}(4)}{=} Z(\widetilde{\mathbf{H}})(\mathbb{F}_p).\end{align*} $$

The extension $1 \rightarrow C \rightarrow E \rightarrow \mathbf {H}(\mathbb {F}_p)' \rightarrow 1$ corresponds to a homomorphism $\alpha :A \rightarrow C$ . In particular, $E=\widetilde {\mathbf {H}}(\mathbb {F}_p) \times C / \delta (A)$ , where $\delta (g)=(\alpha (g),g ^{-1})$ .

Consider A and C as discrete algebraic groups. Let $\mathbf {C}$ be the zero-dimensional algebraic group

$$\begin{align*}\mathbf{C}=\left( C \times Z(\widetilde{\mathbf{H}}) \right)/\delta (A) \end{align*}$$

and let

$$\begin{align*}\mathbf{E}=\left( C \times \widetilde{\mathbf{H}} \right) / \delta(A). \end{align*}$$

The short exact sequence $1 \rightarrow \mathbf {C} \rightarrow \mathbf {E} \rightarrow \mathbf {H} \rightarrow 1$ is a central extension, the map $E = \left( C \times \widetilde {\mathbf {H}}(\mathbb {F}_p)\right) / \delta (A) \rightarrow \mathbf {E}(\mathbb {F}_p)$ is injective and the diagram (1) commutes.

It remains to prove the additional claims. Claim (1) follows from the construction. For Claim (2), assume that $|C|$ is prime to p. By the construction, $\mathbf {E}^\circ =\widetilde {\mathbf {H}}/\ker (\alpha )$ . It follows that $E\cap \mathbf {E}^\circ \supset \widetilde {\mathbf {H}}(\mathbb {F}_p)/\ker (\alpha )$ , so the map $E \cap \mathbf {E}^\circ \rightarrow \mathbf {H}(\mathbb {F}_p)'$ is onto. Since the order of $Z(\widetilde {\mathbf {H}})(\overline {\mathbb {F}_p})$ is prime to p, the same is true for the size of $\mathbf {C}$ and the size of the kernel of $E \cap \mathbf {E}^\circ \rightarrow \mathbf {H}(\mathbb {F}_p)'$ . Since the kernel of the surjection $E \cap\ \mathbf {E}^\circ \rightarrow \mathbf {H}(\mathbb {F}_p)'$ is prime to p, we have that the number of p-elements of $E \cap \mathbf {E}^\circ $ is equal to the number of p-elements of $\mathbf {H}(\mathbb {F}_p)'$ . By the same reasoning applied to the surjection $\mathbf {E}^\circ (\mathbb {F}_p) \rightarrow \mathbf {H}(\mathbb {F}_p)'$ , this is also the number of p-elements of $\mathbf {E}^\circ (\mathbb {F}_p)$ . Hence, all p-elements in $\mathbf {E}^\circ (\mathbb {F}_p)$ are already in $E\cap \mathbf {E}^\circ $ . By Theorem 3.1.1 and Corollary 3.1.3, $\mathbf {E}^\circ (\mathbb {F}_p)'$ is generated by its p-elements, and the second claim follows.

Proof of Proposition 4.0.8.

For every n, let $(\mathcal {R}_n,\mathcal {S}_n,\tau _n)$ be the versal family of reductive groups with involutions from Lemma 3.2.1. By Lemma 3.2.1(3), there is a subscheme $\mathcal {S}^s_n\subset \mathcal {S}_n$ such that, for any geometric point x of $\mathcal {S}_n$ , the group $({\mathcal R}_n)_x$ is absolutely simple adjoint iff x factors through $\mathcal {S}^s_n$ . Let $\mathcal {R}_n^s \subset \mathcal {R}_n$ be the preimage of $\mathcal {S}^s_n$ .

Applying Corollary 4.0.4 to $\mathcal {R}^{s}_n \rightarrow \mathcal {S}^{s}_n$ , we get a family of subgroups $\mathcal {K}_n \Subset \mathcal {R}^{s}_n$ parameterized by an $\mathcal {S}^{s}_n$ -scheme $f:\mathcal {S}_n' \rightarrow \mathcal {S}^{s}_n$ . Let $D(n)$ be the maximum of the number of connected components of a group of the form $(\mathcal {K}_n)_x \cap (\tau _n)_{f(x)}(\mathcal {K}_n)_x$ , or of the form $(\mathcal {K}_n)_x$ , where x is a geometric point of $\mathcal {S}_n'$ .

We define three functions ${C_{LP0}},{C_{LP0}^{ss}},{C_{LP0}^{adj}}:\mathbb {N} \rightarrow \mathbb {N}$ by recursion. Set

$$\begin{align*}{C_{LP0}}(1)={C_{LP0}^{ss}}(1)={C_{LP0}^{adj}}(1)=1 \end{align*}$$

and, for $n \geq 2$ , set

$$\begin{align*}{C_{LP0}^{adj}}(n)=\max \left\{ (3^n+2)^n, {C_{LP0}}(n-1)D(n)^{{n}} \right\}. \end{align*}$$

$$\begin{align*}{C_{LP0}^{ss}}(n)=2^n{C_{LP0}^{adj}}(n). \end{align*}$$

$$\begin{align*}{C_{LP0}}(n)=2^n {C_{LP0}^{ss}}(n). \end{align*}$$

Note that ${C_{LP0}^{adj}} \leq {C_{LP0}^{ss}} \leq {C_{LP0}}$ . We will show that the proposition holds with this choice of ${C_{LP0}}$ . The proof is by induction on $n:={\operatorname {dim}}\,\mathbf {G}$ . The base of the induction, $n=0$ , is trivial. The induction step is divided to the following steps:

1. Step 1: The claim holds if $p \leq 3^{n}+1$ with the bound ${C_{LP0}}$ replaced by ${C_{LP0}^{adj}}$ .

   In this case, we can take $\Delta =1$ , using the bound $|\mathbf {G}(\mathbb {F}_p)| \leq (p+1)^{{\operatorname {dim}}\, \mathbf {G}}$ from [Reference NoriNor87, Lemma 3.5].

2. Step 2: The claim holds if $\mathbf {G}$ is semisimple and adjoint, with the following improvements:

   1. (a) The bound ${C_{LP0}}$ is replaced by ${C_{LP0}^{adj}}$ .

   2. (b) Either $\mathbf {H}$ is semisimple and $Ker(\rho )=1$ , or there is a proper connected t-invariant subgroup $\mathbf {N}<\mathbf {G}$ such that $[\Gamma :\Gamma \cap \mathbf {N}(\mathbb {F}_p)]<D(n)$ .

   By Step 1, we can assume $p>3^{n}+1$ . We have $\mathbf {G}=\mathbf {G}_1 \times \cdots \times \mathbf {G}_m$ , where $\mathbf {G}_i$ are simple and adjoint. We denote the projection $\mathbf {G} \rightarrow \mathbf {G}_i$ by $pr_i$ . Each $\mathbf {G}_i$ is a restriction of scalars from an absolutely simple (and adjoint) group: $\mathbf {G}_i=\operatorname {\mathrm {Res}}_{\mathbb {F}_{q_i}/\mathbb {F}_p} \mathbf {S}_i$ .

   For any $i\in \{1,\dots , m\}$ , we will define a point $\sigma _i\in \mathcal {S}^s_n(\mathbb {F}_{q_i})$ . By Corollary 3.1.8, for any i, we have $t(\mathbf {G}_i)=\mathbf {G}_j$ , for some j. If $i\neq j$ , we take $\sigma _i\in \mathcal {S}^s_n(\mathbb {F}_{q_i})$ to be such that $(\mathcal {R}_n)_{\sigma _i}\cong \mathbf {S}_i$ . This is possible since ${\operatorname {dim}}(\mathbf {S}_i)\leq {\operatorname {dim}}(\mathbf {G}_i)$ . If $i= j$ , we take $\sigma _i=\sigma _j\in \mathcal {S}^s_n(\mathbb {F}_{q_i})$ to be such that $(\mathcal {R}_n)_{\sigma _i}\cong \mathbf {S}_i$ in an $S_2$ -equivariant way.

   In both cases, we identify $\mathbf {G}_i(\mathbb {F}_p)$ , $\mathbf {S}_i(\mathbb {F}_{q_i})$ , and $(\mathcal {R}_n)_{\sigma _i}(\mathbb {F}_{q_i})$ . Given $\Gamma \subset \mathbf {G}(\mathbb {F}_p)$ , there are two cases:

   1. Case 1: For some i, $pr_i(\Gamma ) \subset \mathbf {G}_i(\mathbb {F}_p)$ does not $\mathbb {F}_{q_i}$ -evade $\mathcal {K}_n$ .

      For simplicity, we assume that $t(\mathbf {G}_i)=\mathbf {G}_i$ ; if $t(\mathbf {G}_i)=\mathbf {G}_j$ , the proof is similar (and simpler). In this case, there is a point $x\in {\mathcal S}'(\mathbb {F}_{q_i})$ that lies over $\sigma _i$ such that $pr_i(\Gamma ) \subset (\mathcal {K}_n)_x(\mathbb {F}_{q_i})$ . Denote $\mathbf K:= \mathcal {K}_x$ . By the definition of $D(n)$ , we have $|\pi _0(\mathbf K \cap \mathbf t_{\sigma _i}(\mathbf {K}))|\leq D(n)$ . Let $\mathbf {M}=\operatorname {\mathrm {Res}}_{\mathbb {F}_{q_i}/\mathbb {F}_p}(\mathbf K \cap \mathbf t_{\sigma _i}(\mathbf {K}))$ . We have $|\pi _0(\mathbf M)|\leq D(n)^n$ .

      Using the identification $\mathbf {G}_i=\operatorname {\mathrm {Res}}_{\mathbb {F}_{q_i}/\mathbb {F}_p} \mathbf {S}_i$ , the group $\mathbf {M}$ is a subgroup of $\mathbf {G}_i$ and is defined over $\mathbb {F}_p$ . Note that $pr_i(\Gamma ) \subset \mathbf {M}(\mathbb {F}_p)$ . Since ${\operatorname {dim}}(pr_i ^{-1} (\mathbf {M}))<n$ , the result now follows from the induction step applied to $pr_i ^{-1} (\mathbf {M})^\circ $ and $\Gamma \cap pr_i ^{-1} (\mathbf {M})^\circ (\mathbb {F}_p)$ .

   2. Case 2: For all i, $pr_i(\Gamma )\ \mathbb {F}_{q_i}$ -evades $\mathcal {K}_n$ .

      In this case, there are Frobenius maps $\Phi _i: \mathbf {S}_i(\overline {\mathbb {F}_{q_i}}) \rightarrow \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})$ such that $pr_i(\Gamma )$ is big in $\mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i}$ . Let $\Delta :=\Gamma \cap \prod _i \left( \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i} \right) '$ . Since

      $$\begin{align*}\left[ \mathbf{S}_i(\overline{\mathbb{F}_{q_i}})^{\Phi_i} : \left( \mathbf{S}_i(\overline{\mathbb{F}_{q_i}})^{\Phi_i} \right) '\right] \leq 2^{{\operatorname{dim}}\, \mathbf{S}_i}, \end{align*}$$
      we get
      $$\begin{align*}[ \Gamma : \Delta ] \leq 2^{\sum {\operatorname{dim}}\, \mathbf{S}_i} \leq 2^{{\operatorname{dim}}\, \mathbf{G}}\leq {C_{LP0}^{adj}}(n). \end{align*}$$

      Since $\Delta \supset \Gamma '$ , it follows that $pr_i(\Delta )=\left( \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i} \right) '$ , for all i. Since $\left( \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i}\right)'$ are simple groups, Goursat’s Lemma implies that there is a subset $I \subset [m]$ such that the projection $\Delta \rightarrow \prod _{i\in I} \left( \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i} \right) '$ is an isomorphism. Since $\Delta $ is perfect and $\Delta \subset \Gamma $ , it follows that $\Delta =\Gamma '$ , and, in particular, $t(\Delta )=\Delta $ .

      By [Reference SteinbergSte68, 11.6], there is a connected semisimple $\mathbb {F}_p$ group $\mathbf {H}$ such that $\mathbf {H}(\mathbb {F}_p)\cong \prod _{i\in I} \mathbf {S}_i(\overline {\mathbb {F}_{q_i}})^{\Phi _i}$ .

      By Corollary 3.1.9, the restriction of t to $\Delta $ extends to an involution s of $\mathbf {H}$ .

      By [Reference NoriNor87, Lemma 3.5], $(p-1)^{{\operatorname {dim}}\, \mathbf {H}} \leq \left| \mathbf {H}(\mathbb {F}_p) \right|$ and $\left| \mathbf {G}(\mathbb {F}_p) \right| \leq (p+1)^{{\operatorname {dim}}\, \mathbf {G}}$ , so

      $$\begin{align*}(p-1)^{{\operatorname{dim}}\, \mathbf{H}} \leq \left| \mathbf{H}(\mathbb{F}_p) \right| \leq | \Delta | 2^{{\operatorname{dim}}\,\mathbf{G}} \leq \left| \mathbf{G}(\mathbb{F}_p) \right|2^{{\operatorname{dim}}\, \mathbf{G}} \leq (2p\kern1.5pt{+}\kern1.5pt2)^{{\operatorname{dim}}\, \mathbf{G}} \leq (3p\kern1.5pt{-}\kern1.5pt3)^{{\operatorname{dim}}\, \mathbf{G}}. \end{align*}$$
      Since $p>3^{{\operatorname {dim}}\,\mathbf {G}}+1$ , we have
      $$\begin{align*}\frac{{\operatorname{dim}}\, \mathbf{H}}{{\operatorname{dim}}\,\mathbf{G}} \leq \frac{\log(3p-3)}{\log(p-1)} < 1+\frac{1}{{\operatorname{dim}}\,\mathbf{G}}, \end{align*}$$
      so ${\operatorname {dim}}\,\mathbf {H} \leq {\operatorname {dim}}\,\mathbf {G}$ .

3. Step 3: The claim holds if $\mathbf {G}$ is semisimple (but not necessarily adjoint), with the bound ${C_{LP0}^{ss}}$ .

   Let $\overline {\mathbf {G}}=\mathbf {G}/\mathbf {Z}(\mathbf {G})$ , let $\pi :\mathbf {G} \rightarrow \overline {\mathbf {G}}$ be the projection and let $\overline {\Gamma }=\pi (\Gamma )$ . Applying the previous step to $\overline {\Gamma },\overline {\mathbf {G}}$ , there are two possible cases:

   1. Case 1: There is a proper connected t -invariant subgroup $\overline {\mathbf {N}}<\overline {\mathbf {G}}$ such that $[\overline \Gamma :\overline \Gamma \cap \overline {\mathbf {N}}(\mathbb {F}_p)]<D(n)$ .

   2. Case 2: There is a subgroup $\overline {\Delta } \subset \overline {\Gamma }$ , a semisimple group $\overline {\mathbf {H}}$ with an action of $S_2$ and an $S_2$ -equivariant injective homomorphism $\overline {\rho }:\overline {\Delta } \rightarrow \overline {\mathbf {H}}(\mathbb {F}_p)$ , such that $\overline {\rho }(\overline {\Delta })\subset \overline {\mathbf {H}}(\mathbb {F}_p)$ is big and $[\overline \Gamma :\overline \Delta ]<{C_{LP0}^{adj}}(n)$ .

   In the first case, we are done by the induction assumption. For the second case, note that $\rho \left( \overline \Delta ' \right)=\overline {\mathbf {H}}(\mathbb {F}_p)'$ and, by Lemma 3.1.6, we have $[\overline \Delta :\overline \Delta ']\leq 2^n.$ Denote $\Delta :=\pi ^{-1}(\overline \Delta ') \cap \Gamma $ . We get that

   $$ \begin{align*}[\Gamma:\Delta]<{C_{LP0}^{adj}}(n)2^n={C_{LP0}^{ss}}(n), \end{align*} $$
   where $\Delta $ is a central extension of $\overline {\Delta }'=\overline {\mathbf {H}}(\mathbb {F}_p)'$ . By Lemma 4.0.9, this central extension can be extended to an $S_2$ -equivariant central extension $\mathbf {H}$ of $\overline {\mathbf {H}}$ in such a way that the embedding
   $$ \begin{align*}\Delta \to \mathbf{H}(\mathbb{F}_p)\end{align*} $$
   has big image, as required.

4. Step 4: The claim holds if $\mathbf {G}$ is a direct product of a semisimple group and a torus, with the bound ${C_{LP0}^{ss}}$ .

   Write $\mathbf {G}=\mathbf {G}'\times \mathbf {T}$ . Let $p:\mathbf {G}\to \mathbf {G}'$ be the projection and $\bar \Gamma :=p(\Gamma )$ . Applying the previous step to $\bar \Gamma $ , we get groups $\bar \Delta $ and $\bar {\mathbf {H}}$ . The claim holds for $\Delta :=\bar \Delta \times \mathbf {T}(\mathbb {F}_p)\cap \Gamma $ and ${\mathbf {H}}:=\bar {\mathbf {H}}\times \mathbf {T}$ .

5. Step 5: The claim holds if ${\mathbf {G}}$ is reductive.

   Let $\tilde {\mathbf {G}}:= {\mathbf {G}}' \times Z({\mathbf {G}})^\circ $ . We have an isogeny $\pi :\tilde {\mathbf {G}}\to {\mathbf {G}}$ . By Lemma 3.1.5 and Theorem 3.1.1(5), $[{\mathbf {G}}(\mathbb {F}_p), \pi (\tilde {\mathbf {G}}(\mathbb {F}_p))]\leq 2^n$ . Let $\tilde \Gamma :=\pi ^{-1}(\Gamma )$ . Applying the previous step to $\tilde \Gamma \subset \tilde {\mathbf {G}}(\mathbb {F}_p)$ , we get groups $\tilde \Delta $ , $\tilde {\mathbf {H}}$ and a map $\tilde \rho :\tilde \Delta \to \tilde {\mathbf {H}}(\mathbb {F}_p)$ . Note that $Ker(\pi )\cap \tilde \Delta $ is central in $\tilde \Delta $ , and thus, $\tilde \rho (Ker(\pi )\cap \tilde \Delta )$ is central in $\tilde \rho (\tilde \Delta )$ . Since $\rho (\tilde \Delta )$ is big in $ \tilde {\mathbf {H}}(\mathbb {F}_p)$ , we get (by Lemma 4.0.10) that $\tilde \rho (Ker(\pi )\cap \tilde \Delta )$ is central in $\tilde {\mathbf {H}}$ . Define ${\mathbf {H}}=\tilde {\mathbf {H}}/\tilde \rho (Ker(\pi )\cap \tilde \Delta )$ and $\Delta :=\pi (\tilde \Delta )\cong \tilde \Delta /(Ker(\pi )\cap \tilde \Delta )$ . Note that $[\Gamma :\Delta ]\leq {C_{LP0}}(n)$ . The map $\tilde \rho $ desends to a map $\rho :\Delta \to {\mathbf {H}}(\mathbb {F}_p)$ , and we are done.

6. Step 6: The claim holds for all connected groups ${\mathbf {G}}$ .

   Denoting the unipotent radical of ${\mathbf {G}}$ by $\mathbf {U}$ , t induces an involution on ${\mathbf {G}} / \mathbf {U}$ , and the projection $\pi :{\mathbf {G}} \rightarrow {\mathbf {G}}/\mathbf {U}$ is equivariant. By the previous step, we can assume that $\mathbf {U}$ is positive dimensional. Given $\Gamma $ , let $\bar \Gamma =\pi (\Gamma )$ . By induction, there is a subgroup $\bar \Delta \subset \bar \Gamma $ , an algebraic group $\bar {\mathbf {H}}$ and a homomorphism $\bar \rho : \bar \Delta \rightarrow {\mathbf {H}} (\mathbb {F}_p)$ . It is easy to see that $\Delta := \pi ^{-1} (\bar \Delta )$ , ${\mathbf {H}}:=\bar {\mathbf {H}}$ and $\rho := \bar \rho \circ \pi $ satisfy the requirements of the proposition.

Proof. Suppose that P is a nontrivial normal p-subgroup of $\Gamma $ . Since the index of ${\mathbf {H}}(\mathbb {F}_p)'$ in ${\mathbf {H}}(\mathbb {F}_p)$ is prime to p, we have $P\subset {\mathbf {H}}(\mathbb {F}_p)'$ . For similar reasons, $P\cap Z({\mathbf {H}}(\mathbb {F}_p)')=1$ . This gives an embedding of P into ${\mathbf {H}}(\mathbb {F}_p)'/Z({\mathbf {H}}(\mathbb {F}_p)')$ , which is a product of nonabelian simple groups, a contradiction.

Proof of Corollary 4.0.11.

Set $C_{mon0}(n):=\max (4^n,({C_{LP0}}(n)+1)^n)$ . Suppose that $\Delta \subset \Gamma $ are as in the statement of the corollary. Without loss of generality, we can assume that $\operatorname {\mathrm {Rad}}_p(\Gamma )$ is trivial. Let $n:=\overline {\mathrm {rd}}_p(\Gamma )$ . Using the bound $|{\mathbf {G}}(\mathbb {F}_p)| \leq (p+1)^{{\operatorname {dim}}\, {\mathbf {G}}}$ from [Reference NoriNor87, Lemma 3.5], we may also assume that $p>3$ and $p>{C_{LP0}}(n)$ (otherwise, the claim holds with $\Delta ^\circ =1$ ). Embed $\Gamma \hookrightarrow {\mathbf {G}}(\mathbb {F}_p)$ with ${\mathbf {G}}$ connected reductive of dimension n. Applying Proposition 4.0.8 to $\Delta \subset {\mathbf {G}}(\mathbb {F}_p)$ , there is a normal subgroup $\Delta ^\circ \triangleleft \Delta $ , a connected reductive group ${\mathbf {H}}$ defined over $\mathbb {F}_p$ , and a homomorphism $\rho : \Delta ^\circ \rightarrow {\mathbf {H}}(\mathbb {F}_p)$ such that

1. 1. $[\Delta : \Delta ^\circ ] \leq {C_{LP0}}(n)$ .

2. 2. $\ker \rho $ is a p-group.

3. 3. $\rho (\Delta ^\circ )$ is big in ${\mathbf {H}}(\mathbb {F}_p)$ .

4. 4. ${\operatorname {dim}}\,{\mathbf {H}} \leq n$ .

5. 5. If ${\operatorname {dim}}\,{\mathbf {H}} = n$ , then $\ker \rho =1$ .

By Lemma 4.0.12, $\ker \rho = \operatorname {\mathrm {Rad}}_p(\Delta ^\circ )$ . In particular, $\overline {\mathrm {rd}}_p(\Delta ^\circ ) \leq {\operatorname {dim}}\,{\mathbf {H}}$ .

If ${\operatorname {dim}}\,{\mathbf {H}} < n$ , we are done. Otherwise, since $[\Delta : \Delta ^\circ ] < {C_{LP0}}(n) < p$ , we get that

$$\begin{align*}\operatorname{\mathrm{Rad}}_p(\Delta) \subset \operatorname{\mathrm{Rad}}_p(\Delta ^\circ)=\ker \rho=1, \end{align*}$$

contradicting the condition that $\operatorname {\mathrm {Rad}}_p(\Delta )\neq \Delta \cap \operatorname {\mathrm {Rad}}_p(\Gamma )$ .

Proof. Set ${C_{LP}}(n)={C_{LP0}}(2n).$ Embed $\Gamma \hookrightarrow {\mathbf {G}}$ and apply Proposition 4.0.8 to ${\mathbf {G}} \times {\mathbf {G}}$ , the involution $t(x,y)=(y,x)$ and the subgroup $\left\{ (x,\theta (x)) \mid x\in \Gamma \right\}\cong \Gamma $ .

Proof. We need to show that, for every element $s\in \Omega $ that satisfies $s=\theta (s^{-1})$ , there exists an element $g\in \Omega $ such that $s=\theta (g^{-1})g$ .

Let $g=s^{\frac {|\Omega |+1}{2}}.$ Then

$$ \begin{align*}\theta(g^{-1})g=\theta((s^{\frac{|\Omega|+1}{2}})^{-1}) s^{\frac{|\Omega|+1}{2}}=\theta(s^{-1})^{\frac{|\Omega|+1}{2}} s^{\frac{|\Omega|+1}{2}}=s^{\frac{|\Omega|+1}{2}} s^{\frac{|\Omega|+1}{2}}=s^{|\Omega|+1}=s.\\[-42pt]\end{align*} $$

Proof. Let $g\in \Omega $ . Define $s=(\theta (g^{-1})g)^{\frac {|\Omega |+1}{2}}$ and $o=gs^{-1}$ . Then,

$$ \begin{align*} \theta(s)&=\theta((\theta(g^{-1})g)^{\frac{|\Omega|+1}{2}})\\&= (\theta(\theta(g^{-1})g))^{\frac{|\Omega|+1}{2}}\\&=(g^{-1})\theta(g))^{\frac{|\Omega|+1}{2}}\\&=((\theta(g^{-1}) g)^{-1})^{\frac{|\Omega|+1}{2}}\\&=((\theta(g^{-1}) g)^{\frac{|\Omega|+1}{2}})^{-1})=s^{-1},\\\theta(o)&=\theta(gs^{-1})=\theta(g)s\\&=\theta(g)(\theta(g^{-1})g)^{\frac{|\Omega|+1}{2}}\\&=\theta(g)\theta(g^{-1})g(\theta(g^{-1})g)^{\frac{|\Omega|+1}{2}-1}\\&=g(\theta(g^{-1})g)^{\frac{|\Omega|+1}{2}-1}\\&=g(\theta(g^{-1})g)^{\frac{|\Omega|-1}{2}}\\&=g(\theta(g^{-1})g)^{|\Omega|-\frac{|\Omega|+1}{2}}\\&=g(\theta(g^{-1})g)^{-\frac{|\Omega|+1}{2}}=gs^{-1}=o \end{align*} $$

and, thus,

$$ \begin{align*}g=gs^{-1}s=os.\\[-38pt]\end{align*} $$

Proof. The claims follow from Lemma 5.0.3 and [Reference AizenbudAiz, Theorem 2] (which is an adaptation of results from [Reference Gel’fand and KazhdanGK75] and [Reference Jacquet and RallisJR96]).

Proof. Let n be an integer. Let $\Phi _n:{\mathcal R}_n\to {\mathcal S}_n$ and $\tau _n$ be as in Lemma 3.2.1. Let $s_n$ be the anti-automorphism $s_n(g):=\tau _n(g^{-1})$ . Let $C_{H1R0}(n)$ be such that, for any geometric point x of ${\mathcal S}_n$ ,

$$\begin{align*}\left|\pi_0 \left( \Phi_n^{-1}(x)^{s_n} \right)\right |<C_{H1R0}(n). \end{align*}$$

Fix $x\in {\mathcal S}_n(F)$ and let ${\mathbf {G}}:=\Phi _n^{-1}(x)$ . We need to show that

$$ \begin{align*}|H^1(S_2,{\mathbf{G}}(F))|\leq C_{H1R0(n)}.\end{align*} $$

By definition, we have $H^1(S_2,{\mathbf {G}}(F))={\mathbf {G}}(F)^{s_n}/{\mathbf {G}}(F)$ , where the action of ${\mathbf {G}}$ on ${\mathbf {G}}^{s_n}$ is given by $g \cdot x:=gx s_n(g).$ Since ${\mathbf {G}}$ is connected, Lang’s theorem implies that the map ${\mathbf {G}}(F)^{s_n}/{\mathbf {G}}(F)\to ({\mathbf {G}}^{s_n}/{\mathbf {G}})(F)$ is an embedding. By analyzing the action of the Lie algebra, it is easy to see that the orbits of the action of ${\mathbf {G}}$ on ${\mathbf {G}}^{s_n}$ are open. Thus,

$$ \begin{align*}|H^1(S_2,{\mathbf{G}}(F))|=|{\mathbf{G}}(F)^{s_n}/{\mathbf{G}}(F)|\leq |({\mathbf{G}}^{s_n}/{\mathbf{G}})(F)|\leq |({\mathbf{G}}^{s_n}/{\mathbf{G}})(\overline{F})|=|\pi_0({\mathbf{G}}_{\overline{F}}^{s_n})|\leq C_{H1R0}(n).\end{align*} $$

Proof. Let $\theta \in \operatorname {\mathrm {Aut}}(A)$ be the nontrivial element of $S_2$ . Denote $C=\left\{ x\in A \mid x \theta (x)=1 \right\}$ , $B=\left\{ x ^{-1} \theta (x) \mid x\in A \right\}$ and $S=\left\{ x^2 \mid x\in A \right\}$ . We need to show that $|C/B| \leq |A[2]|^2$ . Since $|C/B| = |C/C\cap S| \cdot |C \cap S / B\cap S| \leq |A[2]|\cdot |C \cap S / B\cap S|$ , it in enough to show that $|C \cap S / B\cap S| \leq |A[2]|$ .

Let $x\in C\cap S$ and let $z\in A$ be such that $z^2=x$ . Then $z \theta (z)\in A[2]$ and $x \left( z ^{-1} \theta (z) \right) = z \theta (z)\in A[2]$ . Hence, $C\cap S \subseteq A[2] \cdot B$ , so $|C \cap S / B\cap S| \leq |A[2] \cdot B / B| \leq |A[2]|$ .

Proof. In the proof, if G is a group with involution $\sigma $ , we identify $H^1(S_2,G)$ with the quotient of $C_{G,\sigma }:=\left\{ x\in G \mid x \sigma (x)=1 \right\}$ by the action of G given by $g \cdot x= \sigma (g)x g ^{-1}$ .

1. 1. We have have a map $\alpha :C_{N,\theta } / N \rightarrow C_{\Gamma ,\theta } / \Gamma $ taking $N \cdot x$ to $\Gamma \cdot x$ . Given $x\in C_{N,\theta }$ , the fiber $\alpha ^{-1} (\alpha (N \cdot x))=\alpha ^{-1} (\Gamma \cdot x)$ is equal to $(\Gamma \cdot x) / N$ , so its size is at most $[\Gamma : N]$ . Hence, $|C_{N,\theta } / N| \leq |C_{\Gamma ,\theta } / \Gamma | \cdot [\Gamma : N]$ .

2. 2. We have a map $\beta :C_{\Gamma ,\theta }/\Gamma \rightarrow C_{\Gamma /N,\theta }/ (\Gamma /N) $ sending $\Gamma \cdot x$ to $(\Gamma / N) \cdot (xN)$ . It is enough to show that the sizes of the fibers of $\beta $ are bounded by $\max _{\tau }|H_\tau ^1(S_2,N)|$ .

   For $x\in C_{\Gamma ,\theta }$ , let $\tau _x:N \rightarrow N$ be the automorphism $\tau _x(n)=x ^{-1} \theta (n) x$ . Since $\tau _x^2(n)=x ^{-1} \theta (x ^{-1}) n \theta (x)x=n$ , the automorphism $\tau _x$ is an involution.

   Suppose that $y\in C_{\Gamma ,\theta }$ and $\beta (\Gamma \cdot x)=\beta (\Gamma \cdot y)$ . Then there is $y'\in \Gamma \cdot y$ such that $y'=x n$ , for some $n\in N$ . Since

   $$ \begin{align*}1=\theta(y')y'=\theta(x)\theta(n)x n=\theta(x) x x ^{-1} \theta(n) x n= \tau_x(n)n,\end{align*} $$

   we get that $n\in C_{N,\tau _x}$ . Hence, $\beta ^{-1} (\beta (\Gamma \cdot x) )$ can be identified with $xC_{N,\tau _x}/ \Gamma $ . For any $m\in N$ ,

   $$\begin{align*}m \cdot y'=\theta(m) x n m ^{-1} = x \tau_x(m) n m ^{-1}, \end{align*}$$

   so $|\beta ^{-1} (\beta (\Gamma \cdot x) )| \leq |xC_{N,\tau _x} / N |=|H^1_{\tau _x}(S_2,N)|$ .

Proof. Take $C_{H1R}(n)=C_{H1R0}({\operatorname {dim}} {\mathbf {G}})4^{{\operatorname {dim}} G}$ , where $C_{H1R0}$ is the function given by Lemma 6.0.1.

Using the bound $|{\mathbf {G}}(\mathbb {F}_p)| \leq (p+1)^{{\operatorname {dim}}\, {\mathbf {G}}}$ from [Reference NoriNor87, Lemma 3.5], we can assume that $p>3$ .

Let $\bar {\mathbf {G}}:={\mathbf {G}}/Z({\mathbf {G}})$ . By Corollary 3.1.3, the induced map ${\mathbf {G}}(\mathbb {F}_p)'\to \bar {\mathbf {G}}(\mathbb {F}_p)'$ is onto, and its kernel is $Z({\mathbf {G}}(\mathbb {F}_p)')$ . Let $\bar \theta $ be the involution induced by $\theta $ on $\bar {\mathbf {G}}(\mathbb {F}_p)'$ . By Theorem 3.1.10, we can lift $\bar \theta $ to an involution t of $\bar {\mathbf {G}}$ . By Lemmas 6.0.3, 6.0.1, 3.1.6 and Theorem 3.1.1, we have

$$ \begin{align*} |H^1(S_2,{\mathbf{G}}(\mathbb{F}_p)')|& \overset{{6.0.3}}{\leq} |H^1(S_2,\bar {\mathbf{G}}(\mathbb{F}_p)')| \max_\tau |H^1_\tau(S_2,Z({\mathbf{G}}(\mathbb{F}_p)'))| {\leq} |H^1(S_2,\bar {\mathbf{G}}(\mathbb{F}_p)')| \cdot |Z({\mathbf{G}}(\mathbb{F}_p)')|\\& \overset{{3.1.1}}{\leq} |H^1(S_2,\bar {\mathbf{G}}(\mathbb{F}_p)')| 2^{{\operatorname{dim}} G} \overset{{6.0.3}}{\leq} |H^1_t(S_2,\bar {\mathbf{G}}(\mathbb{F}_p))|\cdot [\bar {\mathbf{G}}(\mathbb{F}_p):\bar {\mathbf{G}}(\mathbb{F}_p)'] 2^{{\operatorname{dim}} G}\\& \overset{{3.1.6}}{\leq} |H^1_t(S_2,\bar {\mathbf{G}}(\mathbb{F}_p))|\cdot 4^{{\operatorname{dim}} G} \overset{{6.0.1}}{<} C_{H1R0}({\operatorname{dim}} {\mathbf{G}})4^{{\operatorname{dim}} G}=C_{H1R}({\operatorname{dim}} {\mathbf{G}}).\\[-42pt] \end{align*} $$

Proof. We take

$$ \begin{align*}C_{H1}(n):=C_{H1R}(2n){C_{LP}}(n)2^{8n},\end{align*} $$

where $C_{H1R}$ is the function given by Corollary 6.0.4 and ${C_{LP}}$ is the function given by Corollary 4.0.13.

Using the bound $|{\mathbf {G}}(\mathbb {F}_p)| \leq (p+1)^{{\operatorname {dim}}\, {\mathbf {G}}}$ from [Reference NoriNor87, Lemma 3.5] and Lemmas 5.0.2, 6.0.3, we may assume that $p>3.$

Let $\bar \theta $ be the involution of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$ induced by $\theta $ . By Corollary 4.0.13, there is a $\bar \theta $ -invariant normal subgroup $\Delta $ of $\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma )$ , a connected reductive group ${\mathbf {H}}$ defined over $\mathbb {F}_p$ and an involution t of ${\mathbf {H}}$ such that

• • $[\Gamma /\operatorname {\mathrm {Rad}}_p(\Gamma ):\Delta ]\leq C_{LP}(\overline {\mathrm {rd}}(\Gamma ))$ ,

• • ${\operatorname {dim}}\, {\mathbf {H}}\leq 2\overline {\mathrm {rd}}(\Gamma )$ ,

• • There is an equivariant embedding $(\Delta ,\bar \theta |_\Delta )\hookrightarrow ({\mathbf {H}}(\mathbb {F}_p),t)$ ,

• • ${\mathbf {H}}(\mathbb {F}_p)'<\Delta < {\mathbf {H}}(\mathbb {F}_p)$ .

By Corollary 3.1.4, we have $\Delta '={\mathbf {H}}(\mathbb {F}_p)'$ .

Applying Theorem 3.1.1, Corollary 6.0.4 and Lemmas 5.0.2, 6.0.2, 6.0.3, we get

$$ \begin{align*} |H^1(S_2,\Gamma)| &\overset{{6.0.3}}{\leq} |H^1(S_2,\Gamma/\operatorname{\mathrm{Rad}}_p(\Gamma))| \cdot \max_{\tau} |H^1_\tau(S_2,\operatorname{\mathrm{Rad}}_p(\Gamma))|\overset{{5.0.2} }{=} |H^1(S_2,\Gamma/\operatorname{\mathrm{Rad}}_p(\Gamma))| \\&\overset{{6.0.3}}{\leq} \max_{\tau}|H^1_\tau(S_2,\Delta)| \cdot |H^1(S_2,(\Gamma/(\operatorname{\mathrm{Rad}}_p(\Gamma))/\Delta))| \\&\overset{{6.0.3}}{\leq} \max_{\tau} |H^1_{\tau}(S_2,\Delta)| \cdot [\Gamma/\operatorname{\mathrm{Rad}}_p(\Gamma):\Delta] \\&\overset{\quad\,\,\,\,\,}{\leq} \max_{\tau}|H^1_\tau(S_2,\Delta)| C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\& \overset{{6.0.3}}{\leq} \max_{\sigma} |H^1_\sigma(S_2,\Delta')| \cdot \max_{\tau} |H^1_\tau(S_2,\Delta / \Delta')| \cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\& \overset{{6.0.2}}{\leq} \max_{\sigma} |H^1_\sigma(S_2,{\mathbf{H}}(\mathbb{F}_p)')| \cdot | \left( \Delta / \Delta' \right)[2]|^2 \cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\& \overset{{6.0.4}}{\leq} C_{H1R}(2\overline{\mathrm{rd}}(\Gamma)) \cdot | \left( \Delta / \Delta' \right)[2]|^2 \cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\&\overset{\quad\,\,\,\,\,}{\leq} C_{H1R}(2\overline{\mathrm{rd}}(\Gamma)) \cdot | \left( {\mathbf{H}}(\mathbb{F}_p) / {\mathbf{H}}(\mathbb{F}_p)' \right)[2]|^2 \cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\&\overset{\quad\,\,\,\,\,}{\leq} C_{H1R}(2\overline{\mathrm{rd}}(\Gamma)) \cdot | \left( {\mathbf{H}}(\mathbb{F}_p) / {\mathbf{H}}'(\mathbb{F}_p) \right)[2]|^2 \cdot [ {\mathbf{H}}'(\mathbb{F}_p) : {\mathbf{H}}(\mathbb{F}_p)' ]^2\cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\&\overset{{3.1.6}}{\leq} C_{H1R}(2\overline{\mathrm{rd}}(\Gamma)) \cdot | \left( ({\mathbf{H}}/{\mathbf{H}}')(\mathbb{F}_p)\right)[2]|^2 \cdot 2^{4\overline{\mathrm{rd}}(\Gamma)}\cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma)) \\&\overset{\quad\,\,\,\,\,}{\leq} C_{H1R}(2\overline{\mathrm{rd}}(\Gamma)) \cdot 2^{8\overline{\mathrm{rd}}(\Gamma)}\cdot C_{LP}(\overline{\mathrm{rd}}(\Gamma))=C_{H1}(\overline{\mathrm{rd}}(\Gamma)). \\[-39pt]\end{align*} $$

Proof. Define

$$ \begin{align*}C^{her}_{H1}(n):=C_{mon}(n) C_{H1}(n).\end{align*} $$

By Corollary 4.1.2, there is a $\theta $ -invariant normal subgroup $\Delta ^\circ \lhd \Delta $ such that $\overline {\mathrm {rd}}_p(\Delta ^\circ )\leq \overline {\mathrm {rd}}_p(\Gamma )$ and $[\Delta :\Delta ^\circ ]\leq C_{mon}(\overline {\mathrm {rd}}_p(\Gamma ))$ . The previous proposition (Proposition 6.0.5) implies that $|H^1(S_2,\Delta ^\circ )|< C_{H1}(\overline {\mathrm {rd}}_p(\Gamma )).$ From the exact sequence

$$ \begin{align*}H^1(S_2,\Delta^\circ)\to H^1(S_2,\Delta)\to H^1(S_2,\Delta/\Delta^\circ),\end{align*} $$

we get

$$ \begin{align*} |H^1(S_2,\Delta)| &\leq |H^1(S_2,\Delta^\circ)|\cdot |H^1(S_2,\Delta/\Delta^\circ)| \\&\leq C_{H1}(\overline{\mathrm{rd}}_p(\Gamma)) [\Delta:\Delta^\circ] \\&\leq C_{H1}(\overline{\mathrm{rd}}_p(\Gamma)) C_{mon}(\overline{\mathrm{rd}}_p(\Gamma)) \\&= C^{her}_{H1}(\overline{\mathrm{rd}}_p(\Gamma)).\\[-39pt] \end{align*} $$

Proof. For a pair finite group $\Gamma \lhd \Gamma '$ , let $I^{\Gamma '}_{\Gamma }:{\mathcal M}_{\mathbb {F}_p}(\Gamma ') \to {\mathcal M}(\Gamma '/\Gamma )$ be the functor of $\Gamma $ -invariants from the category of $\mathbb {F}_p$ -representations of $\Gamma '$ to the category of $\mathbb {F}_p$ -representations of $\Gamma '/\Gamma $ . Note that, if $p \nmid |\Gamma |$ , then $I^{\Gamma '}_{\Gamma }$ is exact. Also denote

$$ \begin{align*}I_{\Gamma}:=I^{\Gamma}_{\Gamma}.\end{align*} $$

Now

1. 1. If $p \nmid | \Gamma _1 |$ , then

   $$ \begin{align*} H^i(\Gamma_2,\mathbb{F}_p) & \cong R^i(I_{\Gamma_2})(\mathbb{F}_p) \cong R^i(I^{}_{\Gamma_3} \circ I^{\Gamma_2}_{\Gamma_1})(F_p) \cong R^i(I^{}_{\Gamma_3}) \circ I^{\Gamma_2}_{\Gamma_1} (\mathbb{F}_p) \cong \\&\cong R^i(I^{}_{\Gamma_3}) (\mathbb{F}_p) \cong H^i(\Gamma_3,\mathbb{F}_p). \end{align*} $$

2. 2. If $p\nmid | \Gamma _3 |$ , then

   $$ \begin{align*}H^i(\Gamma_2,\mathbb{F}_p)&\cong R^i(I_{\Gamma_2})(\mathbb{F}_p)\cong R^i(I^{}_{\Gamma_3} \circ I^{\Gamma_2}_{\Gamma_1})(F_p)\\&\cong I^{}_{\Gamma_3} \circ R^i I^{\Gamma_2}_{\Gamma_1} (\mathbb{F}_p) \cong I^{}_{\Gamma_3} (H^i(\Gamma_1,\mathbb{F}_p)) \cong H^i(\Gamma_1,\mathbb{F}_p)^{\Gamma_3}.\\[-38pt] \end{align*} $$

Proof. It is enough to show the claim after replacing by $\mathbb {F}_p$ . Define

$$ \begin{align*}C_{H2}(n):=\max(3,{C_{LP0}}(n),4^{n}).\end{align*} $$

Fix n. Let $p>C_{H2}(n)$ be a prime and $\Gamma $ be a finite group such that $\operatorname {\mathrm {Rad}}_p(\Gamma )=1$ , and $\mathrm {rd}_p(\Gamma )\leq n$ . We have to show that

$$ \begin{align*}H^2(\Gamma,\mathbb{F}_p)=0.\end{align*} $$

By Corollary 4.0.13 applied with trivial involution,Footnote ¹ there are a normal subgroup $\Delta \lhd \Gamma $ , a connected reductive algebraic group ${\mathbf {H}}$ defined over $\mathbb {F}_p$ and an injective homomorphism $\rho : \Delta \rightarrow {\mathbf {H}}(\mathbb {F}_p)$ such that

1. 1. $[\Gamma :\Delta ] \leq {C_{LP}}(n)$ .

2. 2. ${\mathbf {H}}(\mathbb {F}_p)'< \rho (\Delta ) < {\mathbf {H}}(\mathbb {F}_p)$ .

3. 3. ${\operatorname {dim}} {\mathbf {H}}\leq 2n$ .

Applying 7.0.1(2) to the embedding $\Delta \subset \Gamma $ , it is enough to prove that $H^2(\Delta ,\mathbb {F}_p)=0$ . We will identify $\rho (\Delta )$ with $\Delta $ .

Let $\Delta _0=\Delta \cap {\mathbf {H}}'(\mathbb {F}_p)$ . We have an embedding of $\Delta /\Delta _0$ into $({\mathbf {H}}/{\mathbf {H}}')(\mathbb {F}_p)$ . Thus, $p \nmid [\Delta ,\Delta _0]$ . By descent of $H^2$ to a subgroup (Lemma 7.0.1(2)), this implies that it is enough to show that

$$ \begin{align*}H^2(\Delta_0,\mathbb{F}_p)=0.\end{align*} $$

By Lemma 3.1.6, $[\Delta _0:{\mathbf {H}}'(\mathbb {F}_p)'] \leq 4^n$ . By Lemma 7.0.1(2), it is enough to show that $H^2({\mathbf {H}}'(\mathbb {F}_p)',\mathbb {F}_p)=0$ .

Let $\pi : \widetilde {{\mathbf {H}}'} \to {\mathbf {H}}'$ the universal cover. By Corollary 3.1.3,

$$\begin{align*}H^2({\mathbf{H}}'(\mathbb{F}_p)',\mathbb{F}_p)=H^2(\pi(\widetilde{{\mathbf{H}}'}(\mathbb{F}_p)),\mathbb{F}_p), \end{align*}$$

and the latter group vanishes by combining descent of $H^2$ to a quotient (Lemma 7.0.1(1)) and vanishing of $H^2$ for simply connected groups (Theorem 3.1.1(2)).

Proof. The map $\phi : {\mathbf {H}}' \times Z({\mathbf {H}}) \rightarrow {\mathbf {H}}$ is an isogeny. By Lemma 3.1.5 and Theorem 3.1.1,

$$\begin{align*}\left[ {\mathbf{H}}(\mathbb{F}_p) : {\mathbf{H}}'(\mathbb{F}_p) \cdot Z({\mathbf{H}})(\mathbb{F}_p) \right] \leq |(\ker \phi)(\overline{\mathbb{F}_p})|=|Z({\mathbf{H}}'(\overline{\mathbb{F}_p})| \leq |Z(\widetilde{{\mathbf{H}}'}(\overline{\mathbb{F}_p}))|\leq 2^{{\operatorname{dim}}\, {\mathbf{H}}'}, \end{align*}$$

where $\widetilde {{\mathbf {H}}'}$ is the universal cover of ${\mathbf {H}}'$ . By Lemma 3.1.6, $[{\mathbf {H}}'(\mathbb {F}_p):{\mathbf {H}}(\mathbb {F}_p)'] \leq 2^{{\operatorname {dim}}\, {\mathbf {H}}'}$ , and the result follows.

Proof. Set

$$ \begin{align*}C_{nr}(n):={C_{LP}}(n)16^{n}C_{rd}(2n).\end{align*} $$

Let $\theta $ be an involution of $\Gamma $ . Applying Corollary 4.0.13 to $\Gamma $ , we get a triple $\Delta ,{\mathbf {H}},t$ . By Lemma 8.0.2, we have

$$\begin{align*}\mu(\Gamma,\theta) \leq [\Gamma:\Delta] \cdot \mu(\Delta,\theta) \leq {C_{LP}}(\mathrm{rd}_p(\Gamma))\mu(\Delta,\theta). \end{align*}$$

Denote $S={\mathbf {H}}(\mathbb {F}_p)'$ and $Z=Z({\mathbf {H}})(\mathbb {F}_p)$ . Note that S and Z are t-invariant subgroups of ${\mathbf {H}}(\mathbb {F}_p)$ . By Lemma 8.0.3,

$$\begin{align*}\mu(\Delta,\theta)\leq [\Delta:\Delta \cap S \cdot Z] \mu(\Delta \cap S \cdot Z,\theta) \leq 2^{{\operatorname{dim}}\, {\mathbf{H}}} \mu (\Delta \cap S \cdot Z,\theta) \leq 4^{\mathrm{rd}_p(\Gamma)} \mu (\Delta \cap S \cdot Z,\theta). \end{align*}$$

Let $\varphi :S \times Z \rightarrow H(\mathbb {F}_p)$ be the multiplication map. $\varphi $ is equivariant if we use the involution $\theta \times \theta $ on $S \times Z$ . By Lemma 8.0.2(1),

$$\begin{align*}\mu(\Delta \cap S \cdot Z,\theta) \leq \mu(\varphi ^{-1} (\Delta), \theta \times \theta). \end{align*}$$

Since $S \subset \Delta $ , we get that $\varphi ^{-1} (\Delta )=S \times A$ , for some $A \subset Z$ . Therefore, by Lemma 8.0.4,

$$\begin{align*}\mu(\varphi ^{-1} (\Delta),\theta \times \theta) \leq \mu(S,\theta). \end{align*}$$

Finally, by Lemma 3.1.6,

$$\begin{align*}\mu(S,\theta) \leq \left[ {\mathbf{H}}'(\mathbb{F}_p):S\right]\mu({\mathbf{H}}'(\mathbb{F}_p),t) \leq 2^{{\operatorname{dim}}({\mathbf{H}}')} \mu({\mathbf{H}}'(\mathbb{F}_p),t)\leq 4^{\mathrm{rd}_p(\Gamma)} \mu({\mathbf{H}}'(\mathbb{F}_p),t), \end{align*}$$

and the result follows from Theorem 8.0.1.

Proof. Write $\rho \restriction _N= \tau ^{\oplus C}$ , where $(\tau ,V) \in Irr(N)$ . Recall the construction of the 2-cocycle corresponding to the triple $(\Gamma ,N,\tau )$ : choose a set of coset representatives $T \subset \Gamma $ such that $1\in T$ . For every $t\in T\smallsetminus \left\{ 1 \right\}$ , choose an isomorphism $A_t: \tau ^t \rightarrow \tau $ such that $\det (A_t)=1$ , and let $A_1=I$ .

Define a map $\pi :\Gamma \to \operatorname {\mathrm {End}}_{\mathbb C}(V)$ by $\pi (tn)=A_t \tau (n)$ for $t\in T$ and $n\in N$ . This is a projective representation of $\Gamma $ that extands $\tau $ and satisfies $\pi (tn)=\pi (t) \pi (n)$ and

$$ \begin{align*}\pi(nt)=\pi(t t^{-1}nt)=A_t \tau(t^{-1}nt)=A_t\tau^{t}(n)=\tau(n) A_t=\pi(n) \pi(t).\end{align*} $$

For every $\gamma _1,\gamma _2\in \Gamma $ , the map

$$ \begin{align*}\pi(\gamma _2 ^{-1})\pi(\gamma_1 ^{-1}) \pi(\gamma_1 \gamma_2 )\end{align*} $$

is an intertwiner of $\tau $ , so

$$ \begin{align*}\pi(\gamma _2 ^{-1})\pi(\gamma _1 ^{-1}) \pi( \gamma _1 \gamma _2 ) = \alpha(\gamma _1,\gamma _2) I,\end{align*} $$

for some $\alpha (\gamma _1,\gamma _2)\in \mathbb {C} ^ \times $ . The map $\alpha :\Gamma \times \Gamma \to \mathbb C^\times $ is a 2-cocycle of $\Gamma $ with coefficients in $\mathbb {C} ^ \times $ , and a simple computation shows that $\alpha $ descends to a 2-cocycle on $\Gamma / N$ . Since N is a p-group, , for every $n\in N$ . Since $\det (A_t)=1$ , for every $t\in T$ , we get that . Since ${\operatorname {dim}}\, \tau $ is a p-th power, we get that .

By assumption, this implies that $\alpha $ is cohomologous to the trivial cocycle, and, therefore, there is a representation $\tau _1$ of $\Gamma $ such that $\tau _1\restriction _{N}=\tau $ . Since $\tau $ is irreducible, so is $\tau _1$ . By [Reference IsaacsIsa06, Corollary 6.17], $\rho =\tau _1 \otimes \tau _2$ , for some $\tau _2\in \operatorname {\mathrm {Irr}}(\Gamma / N)$ .

Proof. Since odd order symmetric pairs are Gelfand pairs (Corollary 5.0.5(1)), we have

$$ \begin{align*}{\operatorname{dim}} \rho_1^{Rad_p(\Gamma)^\theta}\leq 1.\end{align*} $$

If $(\rho _1\otimes \rho _2)^{\Gamma ^\theta }=0$ , the claim trivially holds. Otherwise, let $\chi $ be the character with which $\Gamma ^\theta /Rad_p(\Gamma )^\theta $ acts on $\rho _1^{Rad_p(\Gamma )^\theta }$ . Using Corollary 8.0.5,

$$ \begin{align*}{\operatorname{dim}}(\rho_1\otimes \rho_2)^{\Gamma^\theta}={\operatorname{dim}}(\chi\otimes \rho_2)^{\Gamma^\theta/Rad_p(\Gamma)^\theta}\leq \mu(\Gamma/Rad_p(\Gamma))\leq C_{nr}(\mathrm{rd}(\Gamma/Rad_p(\Gamma)))\leq C_{nr}(\overline{\mathrm{rd}}(\Gamma)).\end{align*} $$

Proof. Define $C_{\nu '}(n)=\max ((C_{H2}(n)+1)^{n},C_{nr}(n))$ . We will prove the corollary by analyzing two cases:

1. Case 1. $p\leq C_{H2}(\overline {\mathrm {rd}}(\Gamma ))$ .

   In this case,

   $$ \begin{align*}|\Gamma/Rad_p(\Gamma)|\leq (p+1)^{\overline{\mathrm{rd}}(\Gamma)}\leq (C_{H2}(\overline{\mathrm{rd}}(\Gamma))+1)^{\overline{\mathrm{rd}}(\Gamma)}\leq C_{\nu'}(\overline{\mathrm{rd}}(\Gamma)).\end{align*} $$

   Since we can control multiplicities when we pass to subgroup of bounded index (Lemma 8.0.2), we get that

   $$ \begin{align*}\nu(\Gamma)\leq |\Gamma/Rad_p(\Gamma)| \cdot \nu(Rad_p(\Gamma)).\end{align*} $$

   Since odd order symmetric pairs are Gelfand pairs (Corollary 5.0.5(1)), we have that

   $$ \begin{align*}\nu(Rad_p(\Gamma))=1.\end{align*} $$

   We obtain

   $$ \begin{align*}\nu^{\prime}_p(\Gamma)\leq \nu(\Gamma)\leq \#(\Gamma/Rad_p(\Gamma)) \leq C_{\nu'}(\overline{\mathrm{rd}}(\Gamma))\end{align*} $$

2. Case 2. $p> C_{H2}(\overline {\mathrm {rd}}(\Gamma ))$ .

   By Proposition 7.0.2, the group is trivial. Let $\rho \in \operatorname {Irr}(\Gamma )$ be such that $\rho |_{Rad_p(\Gamma )}$ is isotypic. By Clifford theory (Lemma 9.0.2), there exist $\pi _1,\pi _2\in \operatorname {Irr}(\Gamma )$ such that $\pi _1|_{Rad_p(\Gamma )}$ is irreducible, the action of $Rad_p(\Gamma )$ on $\pi _2$ is trivial and $\rho \simeq \pi _1\otimes \pi _2$ . Thus, the main theorem for product case (Lemma 10.0.1) implies that, for any involution $\theta $ of $\Gamma $ , we have

   $$ \begin{align*}\rho^{\Gamma^\theta}= (\pi_1\otimes \pi_2)^{\Gamma^\theta}\leq C_{nr}(\overline{\mathrm{rd}}(\Gamma))\leq C_{\nu'}(\overline{\mathrm{rd}}(\Gamma)).\\[-40pt]\end{align*} $$

Proof of Theorem 2.3.1.

Define C recursively by

1. 1. $C(0):=1$

2. 2. $C(n):=C^{her}_{H1}(n)\max (C_{\nu '}(n),C(n-1)C_{mon}(n))$ ,

where $C^{her}_{H1}$ is given by Corollary 6.0.6, $C_{\nu '}$ is given by Corollary 10.0.2 and $C_{mon}$ is given by Corollary 4.1.2. We will prove the theorem by induction. We assume the theorem holds if $\overline {\mathrm {rd}}_p(\Gamma )<n$ and prove it in the case $\overline {\mathrm {rd}}(\Gamma )=n$ .

Let $\theta $ be an involution of $\Gamma $ and $\rho \in \operatorname {\mathrm {Irr}} \Gamma $ . Let $N:=Rad_p(\Gamma )$ . If $(\rho |_N)^{N^\theta }=0$ , we are done. Otherwise, let $\tau $ be an irreducible direct summand of $\rho |_N$ such that $(\tau )^{N^\theta }\neq 0$ . By the Lapid–Prasad property for symmetric pairs of odd order (Corollary 5.0.5(2)), we have $(\tau )^{\theta }\cong \tau ^*$ . Let $\Delta :=\Gamma _\tau $ . We have

$$ \begin{align*}\theta(\Delta)=\Gamma_{\tau^\theta}=\Gamma_{\tau^*}=\Gamma_{\tau}=\Delta,\end{align*} $$

showing that $\Delta $ is $\theta $ -stable.

By Clifford theory (Lemma 9.0.1), we have

$$ \begin{align*}\rho=ind_\Delta^\Gamma(\sigma),\end{align*} $$

where $\sigma \in \operatorname {Irr}(\Delta )$ is such that $\sigma |_N$ is $\tau $ -isotypic. Therefore,

(2) $$ \begin{align} \rho^{\Gamma^\theta}=(ind_\Delta^\Gamma(\sigma))^H= \oplus_{[g]\in \Delta \backslash \Gamma / \Gamma^\theta} \sigma^{\Delta \cap (\Gamma^\theta)^g}= \oplus_{[g]\in \Delta\backslash \Gamma^{dis}/\Gamma^\theta } \sigma^{\Delta \cap (\Gamma^\theta)^g}, \end{align} $$

where

$$ \begin{align*} &\Gamma^{dis} = \{g\in \Gamma| \sigma^{\Delta\cap (\Gamma^\theta)^g} \neq 0\} \subset \{g\in \Gamma| \sigma^{N\cap (\Gamma^\theta)^g} \neq 0\} = \{g\in \Gamma| \tau^{N\cap (\Gamma^\theta)^g} \neq 0\}=\\ &=\{g\in \Gamma| (\tau^g)^{N\cap (\Gamma^\theta)} \neq 0\}=\{g\in \Gamma| (\tau^g)^{N^\theta} \neq 0\}\subset \{g\in \Gamma| (\tau^g)^* \circ \theta \simeq \tau^g\} =:\Gamma^{inv}. \end{align*} $$

The last inclution is by the Lapid-Prasad property for symmetric pairs of odd order (Corollary 5.0.5(2)).

We analyze the quotient $\Delta \backslash \Gamma ^{inv}/\Gamma ^\theta $ . Since $(\tau ^g)^* \circ \theta =(\tau ^*\circ \theta )^{\theta (g)}=\tau ^{\theta (g)}$ , we have

$$ \begin{align*}\Gamma^{inv}=\{g\in \Gamma| \tau^{\theta(g)} \simeq \tau^g\}=\{g| \tau^{g\theta(g^{-1})} \simeq \tau \}=\{g|g\theta(g^{-1})\in\Delta \}.\end{align*} $$

Thus, for any $g\in \Gamma ^{inv}$ , we have

$$ \begin{align*}\theta(ad(g^{-1})(\Delta))=ad(\theta(g^{-1}))(\Delta)= ad(g^{-1}g\theta(g^{-1}))(\Delta)=ad(g^{-1})(\Delta),\end{align*} $$

and hence, $ad(g^{-1})(\Delta )$ is $\theta $ -stable.

We also have

$$ \begin{align*}\Gamma^{inv}/\Gamma^\theta= \{g\theta{g^{-1}\in \Delta|g\in \Gamma}\},\end{align*} $$

and thus,

$$ \begin{align*}\Delta\backslash \Gamma^{inv}/\Gamma^\theta =Ker(H^1(S_2,\Delta)\to H^1(S_2,\Gamma)).\end{align*} $$

Combining the last equality with (2), we get

$$ \begin{align*} {\operatorname{dim}} \rho^{\Gamma^\theta}&\leq |H^1(S_2,\Delta)| \max_{g\in \Gamma^{inv}} {\operatorname{dim}} \sigma^{\Delta \cap (\Gamma^{\theta})^g} \\& = |H^1(S_2,\Delta)| \max\limits_{g\in \Gamma^{inv}} {\operatorname{dim}} (\sigma\circ ad(g))^{(ad(g^{-1})(\Delta)) \cap \Gamma^\theta} \\& = |H^1(S_2,\Delta)| \max\limits_{g\in \Gamma^{inv}} {\operatorname{dim}} (\sigma\circ ad(g))^{(ad(g^{-1})(\Delta))^\theta} \\& = |H^1(S_2,\Delta)| \max\limits_{g\in \Gamma^{inv}} {\operatorname{dim}} \sigma^{\Delta^{ad(g)\circ\theta\circ ad(g^{-1})}} \\& \leq |H^1(S_2,\Delta)|\max\limits_{\theta' \text{ is an involution of } \Delta} ({\operatorname{dim}} (\sigma^{\Delta^{\theta'}})). \end{align*} $$

By our bound on $H^1(S_2,\Delta )$ (Corollary 6.0.6),

(3) $$ \begin{align} {\operatorname{dim}} \rho^{\Gamma^\theta}\leq C_{H1}^{her}(\overline{\mathrm{rd}}_p(\Gamma))\max\limits_{\theta' \text{ is an involution of } \Delta} ({\operatorname{dim}} (\sigma^{\Delta^{\theta'}})). \end{align} $$

We finish the proof by analyzing the following cases:

1. Case 1. $\operatorname {\mathrm {Rad}}_p(\Gamma )\neq \operatorname {\mathrm {Rad}}_p(\Delta )$ :

   In order to bound $\nu (\Delta )$ , we fix an involution $\theta '$ of $\Delta $ . By Corollary 4.1.2, we can find a $\theta '$ -invariant subgroup $\Delta ^\circ \lhd \Delta $ such that

   $$ \begin{align*}\overline{\mathrm{rd}}_p(\Delta^\circ)<\overline{\mathrm{rd}}_p(\Gamma)\end{align*} $$

   and

   $$ \begin{align*}[\Delta:\Delta^\circ] \leq C_{mon}(\overline{\mathrm{rd}}_p(\Gamma)).\end{align*} $$

   The induction assumption implies that $\nu (\Delta ^\circ )\leq C(\overline {\mathrm {rd}}_p(\Gamma )-1)$ . Lemma 8.0.2(2) implies that

   $$ \begin{align*} \nu(\Delta,\theta')&\leq [\Delta:\Delta^\circ]\cdot \nu(\Delta^\circ,\theta'|_{\Delta^\circ}) \leq C_{mon}(\overline{\mathrm{rd}}_p(\Gamma)) C(\overline{\mathrm{rd}}_p(\Gamma)-1), \end{align*} $$

   so

   $$ \begin{align*}\nu(\Delta) \leq C_{mon}(\overline{\mathrm{rd}}_p(\Gamma)) C(\overline{\mathrm{rd}}_p(\Gamma)-1).\end{align*} $$

   It follows that

   $$ \begin{align*} {\operatorname{dim}} \rho^{\Gamma^\theta}\leq C_{H1}^{her}(\overline{\mathrm{rd}}_p(\Gamma ))\cdot C_{mon}(\overline{\mathrm{rd}}_p(\Gamma)) C(\overline{\mathrm{rd}}_p(\Gamma)-1)\leq C(\overline{\mathrm{rd}}_p(\Gamma )),\end{align*} $$
   as required.

2. Case 2. $\operatorname {\mathrm {Rad}}_p(\Gamma )= \operatorname {\mathrm {Rad}}_p(\Delta )$ .

   In this case, $\overline {\mathrm {rd}}_p(\Delta )\leq \overline {\mathrm {rd}}_p(\Gamma )$ , and $\sigma |_{\operatorname {\mathrm {Rad}}_p(\Delta )}$ is isotypic. By (3) and Corollary 10.0.2,

   $$ \begin{align*} {\operatorname{dim}} \rho^{\Gamma^\theta}&\leq C^{her}_{H1}(\overline{\mathrm{rd}}_p(\Gamma))\cdot \nu'(\Delta) \\&\leq C^{her}_{H1}(\overline{\mathrm{rd}}_p(\Gamma)) C_{\nu'}(\overline{\mathrm{rd}}_p(\Delta)) \\&\leq C^{her}_{H1}(\overline{\mathrm{rd}}_p(\Gamma)) C_{\nu'}(\overline{\mathrm{rd}}_p(\Gamma))\leq C(\overline{\mathrm{rd}}_p(\Gamma)).\\[-37pt] \end{align*} $$