1. Introduction
Given a finite nilpotent group
$J$
acting on a finite nilpotent group
$N$
via automorphisms, crossed homomorphisms are maps
$\varphi \,:\, J \to N$
satisfying
$\varphi (\,jj^{\prime}) = \varphi (\,j) \varphi (\,j^{\prime})^{j^{-1}}$
for all
$j,\,j^{\prime} \in J$
. Two such maps
$\varphi$
and
$\varphi ^{\prime}$
are cohomologous if there exists
$n \in N$
such that
$\varphi ^{\prime}(\,j) = n^{-1} \varphi (\,j) n^{j^{-1}}$
for all
$j\in J$
; in this case, we write
$\varphi \sim \varphi ^{\prime}$
. We define the first cohomology
$H^1(J,N)$
to be the pointed set
$Z^1(J,N)$
of crossed homomorphisms modulo this equivalence relation where the distinguished point corresponds to the class containing the map taking each element of
$J$
to the identity of
$N$
.
We first show that the cohomology set
$H^1(J,N)$
decomposes in terms of the first cohomologies of the Sylow
$p$
-subgroups
$J_p$
of
$J$
as follows:
Lemma 1.
For finite nilpotent groups
$J$
and
$N$
, suppose
$J$
acts on
$N$
via automorphisms. Then the map
$\varphi \mapsto \times _{p\in \mathcal {D}} \varphi |_{J_p}$
for
$\varphi \in H^1(J,N)$
induces an isomorphism
$H^1(J,N) \cong \times _{p\in \mathcal {D}} H^1(J_p, N)^{J_p^{\prime}}$
of pointed sets, where
$\mathcal D$
denotes the shared prime divisors of
$\lvert J\rvert$
and
$\lvert N\rvert$
, and for each
$p$
,
$J_p$
is the Sylow
$p$
-subgroup of
$J$
and
$J_p^{\prime}$
is the Hall
$p^{\prime}$
-subgroup of
$J$
.
This parallels the well-known primary decomposition of
$H^1(J,N)$
for abelian
$N$
(see Section 3 for details). As the bijective correspondence between
$H^1(J,N)$
and the
$N$
-conjugacy classes of complements to
$N$
in
$N\rtimes J$
continues to hold for nonabelian
$N$
[Reference Serre6, Exer. 1 in §I.5.1], Lemma1 provides an alternate proof of a result of Losey and Stonehewer [Reference Losey and Stonehewer5]:
Proposition 2 (Losey and Stonehewer). Two nilpotent complements of a normal nilpotent subgroup in a finite group are conjugate if and only if they are locally conjugate.
Here, two subgroups
$H, H^{\prime}\leq G$
are locally conjugate if a Sylow
$p$
-subgroup of
$H$
is conjugate to a Sylow
$p$
-subgroup of
$H^{\prime}$
for each prime
$p$
. It also readily follows that:
Proposition 3.
Let
$G$
be a finite split extension over a nilpotent subgroup
$N$
such that
$G/N$
is nilpotent. If for each prime
$p$
, there is a Sylow
$p$
-subgroup
$S$
of
$G$
such that any two complements of
$S\cap N$
in
$S$
are conjugate in
$G$
, then any two complements of
$N$
in
$G$
are conjugate.
We then establish a fixed point result for nilpotent-by-nilpotent actions in the style of Glauberman:
Theorem 4.
For finite nilpotent groups
$J$
and
$N$
, suppose
$J$
acts on
$N$
via automorphisms and that the induced semidirect product
$N\rtimes J$
acts on some non-empty set
$\Omega$
where the action of
$N$
is transitive. If for each prime
$p$
, a Sylow
$p$
-subgroup of
$J$
fixes an element of
$\Omega$
, then
$J$
fixes an element of
$\Omega$
.
Glauberman showed that this result holds whenever the orders of
$N$
and
$J$
are coprime, without any further restrictions on
$N$
or
$J$
[Reference Glauberman4, Thm. 4]. Thus, this result is only interesting when
$\lvert N\rvert$
and
$\lvert J\rvert$
share one or more prime divisors (i.e. when the action is non-coprime). Analogous results hold if
$N$
is abelian or if
$N$
is nilpotent and
$N\rtimes J$
is supersoluble [Reference Burkhart2].
1.1. Outline
In the remainder of this section, we introduce some notation. We then prove the results in Section 2 and conclude in Section 3.
1.2. Notation
All groups in this note are finite. For a nilpotent group
$J$
, we let
$J_p\in \textrm {Syl}_p(J)$
denote its unique Sylow
$p$
-subgroup and
$J_p^{\prime}$
denote its Hall
$p^{\prime}$
-subgroup so that
$J \cong J_p \times J_p^{\prime}$
. We let
$g^\gamma =\gamma ^{-1}g\gamma$
for
$g,\gamma \in G$
. We otherwise use standard notation from group theory that can be found in Doerk and Hawkes [Reference Doerk and Hawkes3].
For a subgroup
$K\leq J$
, we let
$\varphi |_{K}$
denote the restriction of
$\varphi \in Z^1(J,N)$
to
$K$
and let
$\textrm {res}^J_K \,:\, H^1(J,N) \to H^1(K,N)$
be the map induced in cohomology. For
$\varphi \in Z^1(K,N)$
and
$j\in J$
, define
$\varphi ^j(x) = \varphi (x^{j^{-1}})^{j}$
. We say
$\varphi$
is
$J$
-invariant if
$\textrm {res}^K_{K\cap K^j} \varphi \sim \textrm {res}^{K^j}_{K\cap K^j} \varphi ^j$
for all
$j\in J$
and let
$\textrm {inv}_J H^1(K,N)$
be the set of
$J$
-invariant elements in
$H^1(K,N)$
. For any
$\varphi \in Z^1(J,N)$
, we have
$\varphi ^j(x) = n^{-1} \varphi (x) n^{x^{-1}}$
where
$n= \varphi (\,j^{-1})$
so that
$\varphi ^j \sim \varphi$
. Consequently,
$\textrm {res}^J_K H^1(J,N) \subseteq \textrm {inv}_J H^1(K, N)$
.
For nilpotent
$J$
and
$\varphi \in Z^1(J_p,N)$
, any
$j\in J$
may be written
$j=j_p\times j_p^{\prime}$
for
$j_p \in J_p$
and
$j_p^{\prime} \in J_p^{\prime}$
so that
$\varphi ^j(x) = \varphi (x^{j_p^{-1}})^{j_pj_p^{\prime}} = \varphi ^{\prime}(x)^{j_p^{\prime}}$
for some
$\varphi ^{\prime}\sim \varphi$
. It follows that
$\textrm {inv}_J H^1(J_p, N) = H^1(J_p, N)^{J_p^{\prime}}$
, that is, the
$J$
-invariant elements of
$H^1(J_p, N)$
are those fixed under conjugation by
$J_p^{\prime}$
.
To each complement
$K$
of
$N$
in
$NJ$
, we associate
$\varphi _K\in Z^1(J,N)$
as follows. For
$j\in J$
, we have
$j=n_j^{-1}k_j$
for unique
$n_j\in N$
and
$k_j\in K$
; we then let
$\varphi _K(\,j) = n_j = k_j j^{-1}$
. Conversely, for any
$\varphi \in Z^1(J,N)$
, the subgroup
$F(\varphi ) = \{\varphi (\,j)j\}_{j\in J}$
complements
$N$
in
$NJ$
. In particular,
$F(\varphi _K) = K$
. Furthermore,
$F(\varphi )$
and
$F(\varphi ^{\prime})$
are
$N$
-conjugate in
$NJ$
if and only if
$\varphi \sim \varphi ^{\prime}$
so that
$F$
induces a correspondence between
$H^1(J,N)$
and the
$N$
-conjugacy classes of complements to
$N$
in
$NJ$
. See Serre [Reference Serre6, Ch. I §5] for further details on nonabelian group cohomology.
2. Proofs of results
We begin by establishing Lemma1.
Proof of Lemma 1. As
$N$
is nilpotent, the natural projection maps
$N \to N_p$
induce an isomorphism:
\begin{equation} H^1(J,N) \cong \times _{p\in \mathcal D} H^1(J, N_p), \end{equation}
where terms
$p\notin \mathcal {D}$
drop by the Schur–Zassenhaus theorem [Reference Doerk and Hawkes3, Thm. A.11.3]. Thus, we may focus our attention on
$H^1(J, N_q)$
for some prime
$q$
. If
$J$
is also a
$q$
-group, we are done. Otherwise, we may consider the inflation-restriction exact sequence [Reference Serre6, Sec. I.5.8]:
\begin{equation*} 1 \to H^1(J_q^{\prime}, N_q^{\,J_q}) \to H^1(J,N_q) \xrightarrow {\textrm {res}^J_{J_q}} H^1(J_q,N_q)^{J_q^{\prime}}. \end{equation*}
As
$H^1(J_q^{\prime}, N_q^{\,J_q})$
is trivial,
$\textrm {res}^J_{J_q}$
is injective. For any
$\varphi \in H^1(J_q,N_q)^{J_q^{\prime}}$
, we may define
$\tilde \varphi$
in terms of a representative crossed homomorphism as
$\tilde \varphi (\,j^{\prime}j) = \varphi (\,j)$
for
$j\in J_q$
and
$j^{\prime}\in J_q^{\prime}$
. It is straightforward to verify that
$\tilde \varphi \in Z^1(J, N_q)$
and
$\tilde \varphi |_{J_q} \sim \varphi$
. Thus,
$\textrm {res}^J_{J_q}$
is also surjective and thus an isomorphism. Let
$v\,:\, H^1(J_q, N_q) \to H^1(J_q, N)$
denote the map induced by inclusion. From the decomposition (1),
$H^1(J_q, N_q) \cong H^1(J_q, N)$
, and as
\begin{equation*} H^1(J_q, N_q) \xrightarrow {\nu } H^1(J_q, N) \to H^1(J_q, N_q^{\prime}) \end{equation*}
is exact [Reference Serre6, Prop. I.38] where
$H^1(J_q, N_q^{\prime})$
is trivial, it follows that
$v$
is surjective and hence an isomorphism. As
$\varphi \in H^1(J_p, N_p)^{J_p^{\prime}}$
if and only if
$v(\varphi )\in H^1(J_p, N)^{J_p^{\prime}}$
, it follows that
$\Phi \,:\, H^1(J,N) \to \times _{p\in \mathcal {D}} H^1(J_p, N)^{J_p^{\prime}}$
given by the composition
$\varphi \mapsto \times _{p\in \mathcal {D}} \varphi |_{J_p}$
induces the desired isomorphism:
\begin{equation} H^1(J,N) \cong \textstyle \times _{p\in \mathcal {D}} H^1(J_p, N_p)^{J_p^{\prime}} \cong \textstyle \times _{p\in \mathcal {D}} H^1(J_p, N)^{J_p^{\prime}}. \end{equation}
We now show how Propositions2 and 3 follow from Lemma1.
Proof of Proposition 2. Suppose
$J$
and
$J^{\prime}$
each complement
$N\vartriangleleft G$
as described in the hypotheses of the proposition. Let
$\varphi$
be a crossed homomorphism representing
$J^{\prime}$
in
$H^1(J,N)$
. By hypothesis,
$\varphi |_{J_p}\sim 1|_{J_p}$
for every prime
$p$
, where
$1 \in Z^1(J,N)$
represents the distinguished point. Lemma1 implies
$\varphi \sim 1$
so that
$J$
and
$J^{\prime}$
are conjugate.
Proof of Proposition 3. Suppose
$G \cong N \rtimes J$
satisfies the hypotheses of the proposition. Fix a prime
$p$
. Without loss, we may suppose any two complements of
$N_p$
in
$S=J_pN_p$
are conjugate in
$G$
. If
$J_p^{\prime}$
is such a complement, then
$J_p^{\prime} = (J_p)^g$
for some
$g\in G$
so that
$J_p^{\prime} = (J_p)^{jn} = (J_p)^n$
for some
$j\in J$
and
$n\in N$
. In particular,
$J_p^{\prime}$
is conjugate to
$J_p$
in
$J_pN$
. Thus,
$H^1(J_p,N)$
is trivial. As the choice of prime
$p$
was arbitrary, Lemma1 implies that
$H^1(J,N)$
is also trivial, allowing us to conclude.
To prove Theorem4, we also require:
Proposition 5.
Let
$H$
be a subgroup of
$G\cong N \rtimes J$
where
$N$
and
$J$
are nilpotent. If for each prime
$p$
,
$H$
contains a conjugate of some
$J_p \in \textrm {Syl}_p(J)$
, then
$H$
contains a conjugate of
$J$
.
Proof of Proposition 5. It follows from the hypotheses that
$H$
supplements
$N$
in
$G$
. We induct on the order of
$G$
. If
$H$
is a
$p$
-group or all of
$G$
, the result is immediate. If multiple primes divide
$\lvert N\rvert$
, we have the nontrivial decomposition
$N\cong N_p \times N_p^{\prime}$
for some prime
$p$
. Induction in
$G/N_p$
implies
$J^{n_0} \leq H N_p$
for some
$n_0\in N_p^{\prime}$
. Induction in
$G/N_p^{\prime}$
implies
$J^{n_1} \leq H N_p^{\prime}$
for some
$n_1\in N_p$
. Thus,
$J^{n_0n_1} \leq H N_p \cap HN_p^{\prime} = H$
, where the last equality proceeds from the following argument of Losey and Stonehewer [Reference Losey and Stonehewer5]. Suppose
$g\in H N_p \cap HN_p^{\prime}$
so that
$g=h_0n_0 = h_1n_1$
for some
$h_0, h_1\in H$
,
$n_0\in N_p$
and
$n_1\in N_p^{\prime}$
. Then
$(h_1)^{-1} h_0 = n_1(n_0)^{-1} \in H$
. As
$n_0$
and
$n_1$
commute and have coprime orders, it follows that
$n_0, n_1 \in H$
so
$g\in H$
.
We now proceed under the assumption that
$N=N_q$
for some prime
$q$
. Upon switching to a conjugate of
$H$
if necessary, we may suppose that
$J_q \leq H$
. Let
$Z$
denote the center of
$N$
. If
$Z \cap H$
were nontrivial, then induction in
$G/(Z\cap H)$
would allow us to conclude. Otherwise, in
$G/Z$
, induction implies that
$J^gZ/Z \leq ZH/Z$
for some
$g\in G$
. Let
$\psi \,:\, H \to HZ/Z$
denote the isomorphism between
$H$
and
$HZ/Z$
. Then
$K = \psi ^{-1}(J^gZ/Z)$
complements
$N\cap H$
in
$H$
and
$N$
in
$G$
. Let
$\varphi \in Z^1(K,N)$
correspond to
$J$
. Then
$\varphi |_{K_q}$
corresponds to
$J_q$
where
$[\varphi |_{K_q}]\in H^1(K_q, H \cap N)^{K_q^{\prime}}\cong H^1(K, H \cap N)$
. In particular, there exists a complement, say
$L$
, to
$H\cap N$
in
$H$
that contains
$J_q$
.
$L$
will also complement
$N$
in
$G$
, and as
$\textrm {Syl}_p(L) \subseteq \textrm {Syl}_p(G)$
for all primes
$p\ne q$
, we may apply Proposition2 to conclude that
$J^{g^{\prime}} = L \leq H$
for some
$g^{\prime} \in G$
.
With this, we are prepared to prove Theorem4.
Proof of Theorem 4. Given
$J$
,
$N$
, and
$\Omega$
as described in the hypotheses of the theorem, let
$G= N \rtimes J$
denote the induced semidirect product and consider the stabilizer subgroup
$G_\alpha$
for some
$\alpha \in \Omega$
. As
$N$
acts transitively,
$G=NG_\alpha$
. For each prime
$p$
, the hypotheses of the theorem imply
$(J_p)^{n_p} \leq G_\alpha$
for some
$J_p \in \textrm {Syl}_p(J)$
and
$n_p\in N$
so that Proposition5 allows us to conclude
$J^g \leq G_\alpha$
for some
$g\in G$
. It follows that
$J$
fixes
$g\cdot \alpha$
.
3. Conclusion
We conclude with a brief discussion of analogous results in the abelian case. For arbitrary
$J$
acting on abelian
$N$
, the restriction map
$\textrm {res}^J_{J_p}\,:\, H^1(J,N)_{(p)} \xrightarrow {\cong } \textrm {inv}_J H^1(J_p, N)$
induces an isomorphism for each prime
$p$
, where
$H^1(J,N)_{(p)}$
is the
$p$
-primary component of
$H^1(J,N)$
and
$J_p\in \textrm {Syl}_p(J)$
. Consequently, it follows from the primary decomposition of
$H^1(J,N)$
that [Reference Brown1, Thm. III.10.3]:
\begin{equation} H^1(J,N) \cong \oplus _{p\in \mathcal {D}} \textrm {inv}_J H^1(J_p, N). \end{equation}
Furthermore, for abelian
$N$
, suppose
$G = N \rtimes J$
acts on some non-empty set
$\Omega$
, where the action of
$N$
is transitive, and for each prime
$p$
a Sylow
$p$
-subgroup of
$J$
fixes an element of
$\Omega$
. Then, for arbitrary
$\alpha \in \Omega$
, the stabilizer
$G_\alpha$
splits over
$G_\alpha \cap N$
by Gaschütz’s theorem [Reference Doerk and Hawkes3, Thm. A.11.2] and is locally conjugate and thus conjugate to
$J$
by an argument analogous to the proof of Proposition5. In particular,
$J$
fixes an element of
$\Omega$
. In this note, we find that the decomposition (3) and fixed point result continue to hold for nilpotent
$N$
if
$J$
is also nilpotent.
Acknowledgments
The author thanks the editor C. M. Roney-Dougal and an anonymous reviewer for detailed feedback which considerably improved the manuscript.
Competing interests
The author declares none.
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