A note on noninner automorphisms of order $p$ for finite $p$-groups of coclass 2

Non-inner automorphisms of order p in finite p-groups of coclass 3

In this paper we study the existence of at least one non-inner automorphism of order p of a non-abelian finite p-group of coclass 3, where p is a prime integer such that p is different from 3.

Powerful $p$-groups have noninner automorphisms of order $p$ and some cohomology

In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order $p$ for a finite non-abelian $p$-group. We prove that if $G$ is a finite non-abelian $p$-group such that $G/Z(G)$ is powerful then $G$ has a noninner automorphism of order $p$ leaving either $Φ(G)$ or $Ω_1(Z(G))$ elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd $p$, by showing that the Tate cohomology $H^n(G/N,Z(N))\not=0$ for all $n\geq 0$, where $G$ is a finite $p$-group, $p$ …

有限李型有限群的链接系统的刚性自同构

Rigid automorphisms of linking systems of finite groups of Lie type

研究有限群链接系统中刚性自同构的性质，确定2阶时非内刚性自同构的存在条件，发现例外情况。

A note on non-inner automorphism conjecture

In this paper we prove that every $2$-generator finite $p$-group $G$ has a non-inner automorphism of order $p$ leaving $G^pγ_4(G)$ elementwise fixed ($p\ge 5$). Moreover, we prove a $2$-generator finite $3$-group satisfying $|Ω_1(Z_2(G))|=p^2$ has a non-inner automorphism of order $p$ leaving $G^pγ_3(G)$ elementwise fixed. As a consequence we prove the non-inner automorphism conjecture for every finite $p$-group of coclass $4$ ($p\ge 3$), and coclass $5$ ($p\ge 5$).

Finite p-groups with central automorphism group of minimal order

We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].

Some automorphism groups of finite p-groups

A conjecture of Berkovich asserts that every non-simple finite p-group has a non-inner automorphism of order p. This conjecture is far from being proved despite the great effort devoted to it. In this paper we prove it for p-groups of coclass 2, provided that p is odd. Some related results are also proved, and may be considered as interesting independently.

On groups with a class-preserving outer automorphism

In 1911, Burnside asked whether or not there exist groups that have an outer automorphism which preserves conjugacy classes. Two years later he answered his own question by constructing a family of such groups. Using the small group library in MAGMA we determine all of the groups of order n < 512 that possess such an automorphism. Our investigations led to the discovery of four new infinite families of such groups, all of which are 2-groups of coclass 4.

Co-periodicity isomorphisms between forests of finite p-groups

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which…

On self-similar finite $p$-groups

In this paper, we address the following question: when is a finite $p$-group $G$ self-similar, i.e. when can $G$ be faithfully represented as a self-similar group of automorphisms of the $p$-adic tree? We show that, if $G$ is a self-similar finite $p$-group of rank $r$, then its order is bounded by a function of $p$ and $r$. This applies in particular to finite $p$-groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass $1$, we can fully answer the question above: a $p$-group of maximal class $G$ is self-similar if and only if it contains an elementary…

coclass 1 与 SmallGroup(729,45) 的熵

The entropy of coclass one and of SmallGroup(729,45)

本文首次计算了以 SmallGroup(27,3) 为根的后代树中终顶点的实概率分布熵，并比较了 SmallGroup(729,45) 后代树中实与虚测度的熵。