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The theorem of Aschbacher that is cited by Liebeck, looks at the maximal subgroups $G_0$ of a group $G$ satisfies $SL(a,p^{r/a})\leqslant G \leqslant \Gamma L(a,p^{(r/a})$. For the application of Lieb …

If a permutation group $G$ on a set of size $n$ contains an $n$-cycle then the subgroup $C$ generated by this $n$-cycle is transitive. Thus we have a factorisation $G=CG_\alpha$. Moreover, $C\cap G_\a …

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.

There is a theorem of Fein, Kantor and Schacher that every finite transitive permutation group of degree at least two contains a derangement of prime power order, and hence all the cycle lengths are d …

The Conjecture is still open. Lemma 5 of math.GM/0204209 is false. For example, any primitive group on a prime number of points is a counterexample. Lemma 6 of math/0506617 is also false. Any trans …

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 …

The Boston-Shalev Conjecture asserts that there is a constant $\delta$ such that for any transitive simple group $G$, the proportion of derangments in $G$ is at most $\delta$. After a long sequence of …