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In this paper, we show that all Coleman automorphisms of a finite group with self-central minimal non-trivial characteristic subgroup are inner; therefore the normalizer property holds for these groups. Using our methods we show that the holomorph and wreath product of finite simple groups, among others, have no non-inner Coleman automorphisms. As a further application of our theorems, we provide partial answers to questions raised by M. Hertweck and W. Kimmerle. Furthermore, we characterize the Coleman automorphisms of extensions of a finite nilpotent group by a cyclic $p$-group. Lastly, we note that class-preserving automorphisms of 2-power order of some nilpotent-by-nilpotent groups are inner, extending a result by J. Hai and J. Ge.
Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follow
Let $G$ be a finite group admitting a coprime automorphism $alpha$ of order $e$. Denote by $I_G(alpha)$ the set of commutators $g^{-1}g^alpha$, where $gin G$, and by $[G,alpha]$ the subgroup generated by $I_G(alpha)$. We study the impact of $I_G(alph
Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the b
A finite group $P$ is said to be emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-
Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroup