Proof. By [Reference Li, Bian and Zhang5, Theorem 4.1], $|G:{\mathbf {F}}(G)|\leq {b_m(G)}^{\alpha }$ . By [Reference Isaacs3, Theorem 12.26], there exists an abelian group $B\le {\mathbf {F}}(G)$ such that $|{\mathbf {F}}(G):B|\leq {b_m({\mathbf {F}}(G))}^{4}$ (note that ${b({\mathbf {F}}(G))=b_m({\mathbf {F}}(G))}$ ). Thus,

$$ \begin{align*}|G:B|=|G:{\mathbf{F}}(G)||{\mathbf{F}}(G):B|\leq {b_m(G)}^{\alpha}\cdot b_m({\mathbf{F}}(G))^4.\end{align*} $$

Now, by the Chermak–Delgado theorem [Reference Isaacs4, Theorem 1.41], we conclude that G has a characteristic abelian subgroup A such that

$$ \begin{align*} |G:A|\leq {b_m(G)}^{2\alpha}\cdot b_m({\mathbf{F}}(G))^8.\\[-38pt] \end{align*} $$

Proof. For part (a), we write $\bar {G}=G/{G'}$ . Since $N/(G'\cap N)\cong (NG')/{G'}$ and $\lambda $ can be viewed as a character in $N/(G'\cap N)$ , we have $\bar {\lambda } \in {\operatorname {Irr}}(NG'/{G'})$ . The group $\bar {G}=G/{G'}$ is abelian, so $\bar {\lambda }$ is extendible to $\bar {G}$ and it follows that $\lambda $ is extendible to G.

For part (b), we write

$$ \begin{align*}\chi : {N}\rightarrow {{\mathbb{F}}}^*\end{align*} $$

for a module representation of $\lambda $ which affords $\chi $ . For $g\in G$ , let $g=ah$ , where $a\in N$ and $h\in H$ . We define

$$ \begin{align*}\tilde{\chi}(g)=\chi(a)\in {{\mathbb{F}}}^*.\end{align*} $$

We can calculate directly that

$$ \begin{align*} \tilde{\chi}(ah\cdot a'h')&=\tilde{\chi}(ah\cdot a'\cdot hh^{-1}\cdot h') =\tilde{\chi}(a\cdot ha'h^{-1}\cdot hh')\\ &=\chi(a\cdot ha'h^{-1}) =\chi(a)\cdot \chi (ha'h^{-1})\\ &=\chi(a)\cdot \chi (a') =\tilde{\chi}(ab)\cdot \tilde{\chi}(a'h'). \end{align*} $$

Thus, $\tilde {\chi }$ is an extension of $\chi $ and it follows that $\lambda $ is extendible to G.

Proof. Let $\bar {G} := G/{\mathbf {F}}(G)$ . Since ${\mathbf {F}}(G/\Phi (G))= {\mathbf {F}}(G)/\Phi (G)$ , we may assume that $\Phi (G)=1$ . Thus, ${\mathbf {F}}(G)$ is abelian. Now, G splits over the abelian normal subgroup ${\mathbf {F}}(G)$ . Also, ${\mathbf {F}}(G)$ is a faithful and completely reducible $\bar {G}$ -module by Gaschütz’s theorem [Reference Manz and Wolf6, Theorem 1.12]. By [Reference Manz and Wolf6, Proposition 12.1], ${\operatorname {Irr}}({\mathbf {F}}(G))$ is a faithful and completely reducible $\bar {G}$ -module. Since ${\mathbf {O}}_{p}(G)=1$ , we have ${\operatorname {Irr}}({\mathbf {F}}(G))={\operatorname {IBr}}({\mathbf {F}}(G))$ , and thus ${\operatorname {IBr}}({\mathbf {F}}(G))$ is a faithful and completely reducible $\bar {G}$ -module. By [Reference Yang11, Theorem 3.4], there exists $\beta \in {\operatorname {IBr}}({\mathbf {F}}(G))$ such that $|\bar {G}|\le |{\bar {G}}:{\bar {I}}|^{\alpha }$ , where $\bar {I}=I_{\bar {G}} (\beta )=\{\bar {g}\in \bar {G} \ |\ \beta ^{\bar {g}}=\beta \}$ . Let I be the preimage of $\bar {I}$ in G and $I=I_G (\beta )=\{g\in G \ |\ \beta ^g=\beta \}$ . By Lemma 2.3, let $\widehat {\beta } \in {\operatorname {IBr}} (I|\beta )$ be an extension of $\beta $ and consider $\chi :={\widehat {\beta }}^G\in {\operatorname {IBr}}_m(G)$ . Then,

$$ \begin{align*}|G:{\mathbf{F}}(G)|=|\bar{G}|\le |{\bar{G}}:{\bar {I}}|^{\alpha}=|G:I|^{\alpha}\le {\chi}(1)^{\alpha}\le {b_{\textit{mp}}(G)}^{\alpha}.\end{align*} $$

By [Reference Isaacs3, Theorem 12.26], there exists an abelian group $B\le {\mathbf {F}}(G)$ such that $|{\mathbf {F}}(G):B|\leq {b({\mathbf {F}}(G))}^{4}$ . Since ${\mathbf {O}}_{p}(G)=1$ , we have $b({\mathbf {F}}(G))=b_p({\mathbf {F}}(G))=b_{\textit {mp}}({\mathbf {F}}(G))$ . Thus,

$$ \begin{align*} |G:B|=|G:{\mathbf{F}}(G)||{\mathbf{F}}(G):B|\leq {b_{\textit{mp}}(G)}^{\alpha}\cdot b_{\textit{mp}}({\mathbf{F}}(G))^4. \end{align*} $$

By the Chermak–Delgado theorem [Reference Isaacs4, Theorem 1.41], we conclude that G has a characteristic abelian subgroup A such that

$$ \begin{align*} |G:A|\leq {b_{\textit{mp}}(G)}^{2\alpha}\cdot b_{\textit{mp}}({\mathbf{F}}(G))^8.\\[-38pt] \end{align*} $$

Proof. By Gaschütz’s theorem [Reference Manz and Wolf6, Theorem 1.12], $\bar {G}=G/{\mathbf {F}}(G)$ acts faithfully and completely reducibly on $V={\mathbf {F}}(G)/\Phi (G)$ . By [Reference Manz and Wolf6, Proposition 12.1], ${\operatorname {Irr}}(V)$ is a faithful and completely reducible $\bar {G}$ -module. Since ${\mathbf {O}}_{p}(G)=1$ , we have $p\nmid |{\mathbf {F}}(G)|$ and ${\operatorname {Irr}}(V)={\operatorname {IBr}}(V)$ . Applying [Reference Yang11, Theorem 3.4] to this action, we deduce that there exists ${\lambda \in {\operatorname {IBr}}(V)}$ such that

$$ \begin{align*}|\bar{G}|\leq |G:I_{G}(\lambda)|^{\alpha}.\end{align*} $$

By Clifford’s correspondence [Reference Navarro10, Theorem 8.9], all the characters in ${\operatorname {IBr}}(G|\lambda )$ are induced from irreducible Brauer characters of $I_G(\lambda )$ . In particular, if $\chi \in {\operatorname {IBr}}(G|\lambda )$ , then

$$ \begin{align*}|\bar{G}|\leq |G:I_{G}(\lambda)|^{\alpha}\leq \chi(1)^{\alpha}.\end{align*} $$

It follows that

$$ \begin{align*} |G:{\mathbf{F}}(G)|\leq {\operatorname{acd}}_p(G|\lambda)^{\alpha}.\\[-34pt] \end{align*} $$

Proof. Let P be a Sylow p-subgroup of H and let $F={\mathbf {F}}(H)$ . Set $K=PF$ . We note that K acts faithfully on V, so ${\mathbf {F}}(KV)=V$ . This implies that K acts faithfully and completely reducibly on ${\operatorname {Irr}}(V)$ . By [Reference Dolfi and Navarro1, Theorem 1.1], there exists a P-invariant $\lambda \in {\operatorname {Irr}}(V)$ such that $|F:C_F(\lambda )|\geq \sqrt {|F|}$ . Clifford’s correspondence implies that if $\widehat {\lambda }\in {\operatorname {Irr}}(G|\lambda )$ , then ${\widehat {\lambda }(1)\geq \sqrt {|F|}}$ .

By Lemma 2.7, $\lambda $ can be extended to $C_G(\lambda )$ . Since $P\leq C_G(\lambda )$ , by using Clifford’s correspondence again, all the characters in ${\operatorname {Irr}}(G|\lambda )$ are induced from irreducible characters of $C_G(\lambda )$ . We deduce that there exists a $p'$ -degree character $\widehat {\lambda }\in {\operatorname {Irr}}(G|\lambda )$ . Then, we have a $p'$ -degree character $\chi :=\widehat {\lambda }^{G}\in {\operatorname {Irr}}_m(G)$ and $\chi (1)\geq \sqrt {|F|}$ . Hence,

$$ \begin{align*}\sqrt{|F|}\leq \chi(1)\leq b_{mp'}(G). \end{align*} $$

Thus, we deduce that $|{\mathbf {F}}(H)|\leq b_{mp'}(G)^2$ .

Proof. Without loss of generality, we may assume that ${\mathbf {O}}_{p'}(G)=1$ . We write ${V={\mathbf {O}}_{p}(G)}$ and note that V is the Fitting subgroup G. By Gaschütz’s theorem, we may assume that $\Phi (G)=1$ , so that V is elementary abelian. We write $G=HV$ with ${H\cap V=1}$ and note that H acts faithfully and completely reducibly on V. Let $F={\mathbf {F}}(H)$ . Then, $|{\mathbf {F}}(H)|\leq b_{mp'}(G)^2$ by Lemma 2.8.

From [Reference Manz and Wolf6, Theorem 3.5], $|H:F|\leq |F|^{9/4}$ , and thus

$$ \begin{align*} |H|=|H:F||F|\leq |F|^{13/4}\leq b_{mp'}(G)^{13/2}. \\[-34pt] \end{align*} $$