Volume 26, Issue 2

Let 𝐾 be a finite simple group of Lie type over a field of even order q > 2 q>2 . If 𝐾 is not F 4 2 ⁢ ( q ) {}^{2}F_{4}(q) , then we determine the fusion systems ℱ of J-component type with a fully centralized involution 𝑗 such that C F ⁢ ( j ) C_{\mathcal{F}}(j) has a component realized by 𝐾. The exceptional case is treated in a later paper.

We construct ( 2 , 3 ) (2,3) -generators for the finite 8-dimensional orthogonal groups, proving the following results: the groups Ω 8 + ⁢ ( q ) \Omega_{8}^{+}(q) and P ⁢ Ω 8 + ⁢ ( q ) \mathrm{P}\Omega_{8}^{+}(q) are ( 2 , 3 ) (2,3) -generated if and only if q ≥ 4 q\geq 4 ; the groups Ω 8 - ⁢ ( q ) \Omega_{8}^{-}(q) and P ⁢ Ω 8 - ⁢ ( q ) \mathrm{P}\Omega_{8}^{-}(q) are ( 2 , 3 ) (2,3) -generated for all q ≥ 2 q\geq 2 .

It is known that a finite group with an automorphism 𝜑 of coprime order has a soluble radical of ( | φ | , | C G ⁢ ( φ ) | ) (\lvert\varphi\rvert,\lvert C_{G}(\varphi)\rvert) -bounded Fitting height and index. We extend this classical result as follows. Let f ⁢ ( x ) = a 0 + a 1 ⋅ x + ⋯ + a d ⋅ x d ∈ Z ⁢ [ x ] f(x)=a_{0}+a_{1}\cdot x+\cdots+a_{d}\cdot x^{d}\in\mathbb{Z}[x] be a primitive polynomial, and let 𝐺 be a finite group with an automorphism 𝜑 of coprime order satisfying g a 0 ⋅ φ ⁢ ( g ) a 1 ⁢ ⋯ ⁢ φ d ⁢ ( g ) a d = 1 g^{a_{0}}\cdot\varphi(g)^{a_{1}}\cdots\varphi^{d}(g)^{a_{d}}=1 for all g ∈ G g\in G . Then the soluble radical of 𝐺 has ( d , | C G ⁢ ( φ ) | ) (d,\lvert C_{G}(\varphi)\rvert) -bounded Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree 𝑑.

In this paper, we set η ⁢ ( G ) \eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G ) \eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2 \eta(G)\geq n-2 , and we determine when equality holds.

Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group Out ⁡ ( G ) \operatorname{Out}(G) , or in other words of embeddings of Out ⁡ ( G ) \operatorname{Out}(G) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some 𝑚. This is based on work of Bridson and Vogtmann in which they construct embeddings of Out ⁡ ( F n ) \operatorname{Out}(F_{n}) into Out ⁡ ( F m ) \operatorname{Out}(F_{m}) for some values of 𝑛 and 𝑚 by interpreting Out ⁡ ( F n ) \operatorname{Out}(F_{n}) as the group of homotopy equivalences of a graph 𝑋 of genus 𝑛, and by lifting homotopy equivalences of 𝑋 to a characteristic abelian cover of genus 𝑚. Our construction for a free product 𝐺, using a presentation of Out ⁡ ( G ) \operatorname{Out}(G) due to Fuchs-Rabinovich, is written as an algebraic proof, but it is directly inspired by Bridson and Vogtmann’s topological method and can be interpreted as lifting homotopy equivalences of a graph of groups. For instance, we obtain a faithful free representation of Out ⁡ ( G ) \operatorname{Out}(G) when G = F d ∗ G d + 1 ∗ ⋯ ∗ G n G=F_{d}\ast G_{d+1}\ast\cdots\ast G_{n} , with F d F_{d} free of rank 𝑑 and G i G_{i} finite abelian of order coprime to n - 1 n-1 .