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The maximal subgroups of $A_n$ are given by the O'Nan-Scott Theorem. They lie in one of the following classes: 1) $A_n \cap (S_{n-k} \times S_k)$, that is the stabiliser of a $k$-set. 2) $A_n \cap ( …

Yes this follows from work of Baumann and Thompson. See the paper `On finite insoluble groups with nilpotent maximal subgroups' by John Rose https://doi.org/10.1016/0021-8693(77)90301-5 In fact, the …

You have left out a possbility for a monolithic action. It is possible to have $U\cap N=1$. This can occur when either S is abelian (here G is a subgroup of AGL(d,p) in its usual action on $p^d$ point …

The answer is no. $G=MN$ is an exact factorisation is equivalent to $N$ acting regularly on the set of right cosets of $M$ in $G$. It is not necessary for two regular subgroups of a group to be isomo …

It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion. Is anything known about f …

By a result of Horoševskiĭ you can never find such an automorphism, that is all automorphisms of finite simple groups have a regular orbit.

It is true for all primitive groups: The primitive groups of degree n containing an n-cycle were independently classified in Li, Cai Heng The finite primitive permutation groups containing an abelian …

This is an example of a much more general embedding. Let $q$ be a prime power and $m$ a positive integer. Let $V$ be an $m$-dimensional vector space over $F=GF(q^2)$ and let $B:V\times V\rightarrow F$ …

The maximal subgroups of odd index in finite simple groups were classified in Liebeck and Saxl - The primitive permutation groups of odd degree and independently in Kantor - Primitive permutation grou …

The answer will depend on $q$, $\epsilon$ and potentially also $\ell$. Instead of looking at spinor norms you can find an element $x$ such that $\mathrm{SO}_2^\epsilon(q^\ell)=\langle \Omega_2^\epsilo …

There is a classical result of Wielandt that if you assume the Schreier conjecture (that the outer automorphism group of an finite nonabelian simple groups is solvable), then a group of degree n other …

We looked at this question during our research retreat and obtained the following characterisation: If $H$ is transitive on $X$ then it will be generated by its point stabilisers if and only if it doe …

The proof for this appeared over a series of papers. The final one was Jan Saxl, `On Finite Linear Spaces with Almost Simple Flag-Transitive Automorphism Groups' Journal of Combinatorial Theory, Seri …