Derangements in cosets of primitive permutation groups

Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed-point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O'Nan–Scott Theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

Finite groups all of whose small subgroups are normal

The finite groups G such that, whenever H < G and |H|2 < |G|, then H ◃ G are classified in Theorems 2.1 and 3.2. Let pν(X)(pa(X)) be the maximal order of an abelian (a minimal nonabelian) subgroup of a p-group X. We classify the p-groups G such that, whenever an abelian H < G and |H| ≤ pν(G)-2, then H ◃ G. Next, we describe the p-groups G all of whose abelian subgroups of order ≤ pa(G)-2 are normal. Minimal nonabelian and minimal nonnilpotent groups play crucial role in the whole paper.

FINITE GROUPS IN WHICH ALL NONABELIAN SUBGROUPS ARE TI-SUBGROUPS

In this paper, we show that G is a finite group in which every nonabelian subgroup is a TI-subgroup if and only if every nonabelian subgroup of G is normal in G.