On Automorphisms of Finite $p$-groups

For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|\Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|\Aut(G)|$ for every non-abelian finite $p$-group $G$.

This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite $p$-group is itself a $p$-group. We originally proved this theorem in a paper with Martin; the presentation of the proof here contains omitted proof details and revised exposition. We also give a survey of the extant results on automorphism groups ...

Let $G$ be a finite $p$-group.

Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by $\Cl(a)=\{gag^{-1}\mid g\in G\}$ the conjugacy class of $a$ in $G$. Assume that $|\Cl(a)|=p^n$. Then $\Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in \Cl(a^{-1})\}$ is the union of at least $n(p-1)+1$ distinct conjugacy classes of $G$.

In this paper, we have computed the automorphism groups of all groups of order $p^{2}q^{2}$, where $p$ and $q$ are distinct primes.

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. Let $\mathrm{Aut}_c(G)$ and $\mathrm{Aut}_z(G)$ respectively denote the group of all class preserving and central automorphisms of $G$. We give a necessary condition for $G$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and give necessary and sufficient conditions for $G$ with elementary abelian or cyclic center such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G).$ We also characterize all finite $p$-groups $G$ of ...

Let $\alpha$ be a coprime automorphism of a group $G$ of prime order and let $P$ be an $\alpha$-invariant Sylow $p$-subgroup of $G$. Assume that $p\notin \pi(C_G(\alpha))$. Firstly, we prove that $G$ is $p$-nilpotent if and only if $C_{N_G(P)}(\alpha)$ centralizes $P$. In the case that $G$ is $Sz(2^r)$ and $PSL(2,2^r)$-free where $r=|\alpha|$, we show that $G$ is $p$-closed if and only if $C_G(\alpha)$ normalizes $P$. As a consequences of these two results, we obtain that $G\...

We characterize all finite p-groups G of order p^n(n\leq 6), where p is a prime for n\leq 5 and an odd prime for n = 6, such that the center of the inner automorphism group of G is equal to the group of central automorphisms of G.

Every semidirect product of groups $K\rtimes H$ has size $\left | K \right | \cdot \left | H\right |$, yet the size of such a group's automorphism group varies with the chosen action of $H$ on $K$. This paper will explore groups of the form $\text{Aut}(K\rtimes H)$, considering especially the case where $H$ and $K$ are cyclic. Only finite groups will be considered.

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-length, the number of generators, and p. The proof of...