The multiple holomorph of a semidirect product of groups having coprime exponents

Abstract

Given any group G, the multiple holomorph NHol(G) is the normalizer of the holomorph Hol(G) = ρ(G) ⋊ Aut(G) in the group of all permutations of G, where ρ denotes the right regular representation. The quotient T (G) = NHol(G)/HolG) has order a power of 2 in many of the known cases, but there are exceptions. We shall give a new method of constructing elements (of odd order) in T (G) when G = A ⋊ C d , where A is a group of finite exponent coprime to d and C d is the cyclic group of order d.

Non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension

“…Proof It is clear from Proposition 2.1 and (2.1) that every regular subgroup of $$\mathrm{Hol}(N)$$ isomorphic to $$G$$ is of the stated shape. The converse is also true by the calculation in [17, Proposition 3.4].$$\Box$$…”

“…Proof See [16, Proposition 2.1] for the first claim, and that $$h$$ is a homomorphism was verified in [17, Proposition 3.4]. By (2.1) and the bijectivity of $$g$$, it is clear that $$f(G)h(G)$$ contains $$\mathrm{Inn}(N)$$ and $$f(G)\mathrm{Inn}(N) = h(G)\mathrm{Inn}(N)$$.…”

“…Indeed, suppose that $$$$\begin{equation*} S \leqslant N \leqslant \mathrm{Aut}(S), \end{equation*}$$$$where $$S$$ is a nonabelian simple group identified with $$\mathrm{Inn}(S)$$. Then as is well known (or see [17, Lemma 4.3]), the natural homomorphism $$$$\begin{equation*} \mathrm{Aut}(N)\longrightarrow \mathrm{Aut}(S);\,\ \varphi \mapsto \varphi |_S \end{equation*}$$$$induced by restriction is injective and it maps $$\mathrm{Inn}(N)$$ to $$N$$. It follows that $$P$$ is also embedded between $$S$$ and $$\mathrm{Aut}(S)$$, whence $$P$$ is an almost simple group with socle isomorphic to $$S$$. Since $$G$$ is solvable, its quotients $$A$$ and $$B$$ are also solvable. Almost simple groups that are factorizable as the product of two solvable subgroups have been studied in [12, Proposition 4.1] (its proof is based on a result of [11] and some computations in Magma […”

The multiple holomorph of centerless groups

“…Proof of (12), (13). Let σ, τ ∈ G. First, we know by ( 3) that [f (σ), h(τ )] = 1 and [h(σ), f (τ )] = 1.…”

“…(a) g induces an anti-homomorphism from G to G/ ker(f ). (b) g induces a homomorphism from G to G/ ker(h).In particular, by(12),(13), both (a) and (b) are satisfied if (c) below holds.…”