For a finite abelian p-group A and a subgroup $\Gamma \le \operatorname {\mathrm {Aut}}(A)$, we say that the pair $(\Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${\mathcal {F}}$ over a finite p-group $S\ge A$ such that $C_S(A)=A$, $\operatorname {\mathrm {Aut}}_{{\mathcal {F}}}(A)=\Gamma $ as subgroups of $\operatorname {\mathrm {Aut}}(A)$, and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $\Gamma $ one of the Mathieu groups, that the only ${\mathbb {F}}_p\Gamma $-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all 𝔽pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups.

Let A and G be finite groups and suppose that A acts via automorphisms on G with $(|A|, |G|)=1$. We study how certain conditions on the Sylow 2-subgroups of the fixed point subgroup of the action $C_G(A)$ may imply the non-simplicity or solubility of G.

Let $G$ be a finite group and let $\mathcal{F}$ be a class of groups. Then ${{Z}_{\mathcal{F}\Phi }}\left( G \right)$ is the $\mathcal{F}\Phi$-hypercentre of $G$, which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal{F}$-central in $G$. A subgroup $H$ is called $\mathcal{M}$-supplemented in a finite group $G$ if there exists a subgroup $B$ of $G$ such that $G\,=\,HB\,\text{and}\,{{H}_{1}}B$ is a proper subgroup of $G$ for any maximal subgroup ${{H}_{1}}$ of $H$. The main purpose of this paper is to prove the following: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P\,\text{of}\,{{F}^{*}}\left( E \right)$ has a subgroup $D$ such that $1\,<\,\left| D \right|\,<\left| P \right|$ and every subgroup $H\,\text{of}\,P$ with order $\left| H \right|\,=\,\left| D \right|$ is $\mathcal{M}$-supplemented in $G$, then $E\,\le \,{{Z}_{\mathcal{U}\Phi }}\left( G \right)$.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite group of order $n$, and let $\text {C}_n$ be the cyclic group of order $n$. For $g\in G$, let ${\mathrm{o}}(g)$ denote the order of $g$. Let $\phi $ denote the Euler totient function. We show that $\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$, with equality if and only if $G$ is isomorphic to $\text {C}_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of bidirectional edges in its directed power graph.

We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S3-free saturated 2-fusion systems are constrained. This supplies a new proof of Glauberman's Theorem on S4-free groups and its various corollaries.