Minimal odd order automorphism groups

Outer automorphisms of free Burnside groups

In this paper, we study some properties of the outer automorphism group of free Burnside groups of large odd exponent. In particular, we prove that it contains free and free abelian subgroups.

Automorphism tower problem and semigroup of endomorphisms for free Burnside groups

We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower…

Growth and order of automorphisms of free groups and free Burnside groups

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.

Coleman automorphisms of finite groups and their minimal normal subgroups

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 n…

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。

Non-inner automorphisms of order p in finite normally constrained p-groups

In this paper we study the existence of at least one non-inner automorphism of order p in a finite normally constrained p-group when p is an odd prime.

Absolutely simple polarized abelian varieties of odd Sophie Germain prime dimension over finite fields with maximal automorphism groups

For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the asymptotic behavior of prime numbers that are related to our construction.

关于正则根树自同构群的言语子群

On verbal subgroups of the group of automorphisms of regular rooted trees

本文重新审视了 $d>2$ 且为奇数时正则根树自同构群的所有言语子群的描述。

On central automorphisms of finite p-groups

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.

Automorphisms of nonsplit extensions of 2-groups by $PSL_2(q)$

We complete the description of automorphism groups of all nonsplit extensions of elementary abelian $2$-groups by $PSL_2(q)$, with $q$ odd, for an irreducible induced action. An application of this result to the theory of $π$-submaximal subgroups is given.

Splitting automorphisms of prime power orders of free Burnside groups

We prove that if the order of a splitting automorphism of free Burnside group~$B(m,n)$ of odd period~$n\ge1003$ is a prime power, then the automorphism is inner. Thus, we give an affirmative answer to the question on the coincidence of splitting and inner automorphisms of free Burnside groups $B(m,n)$ for automorphisms of orders~$p^k$ ($p$ is a prime number). This question was posed in the Kourovka Notebook in 1990 (see 11th ed., Question 11.36.~b).

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.

循环群全形的自同构群

Automorphism Group of the Holomorph of a Cyclic Group

本文证明了当阶为 $n$ 的循环群的 $n$ 为奇素数幂的两倍时，该循环群的全形同构于其自身的自同构群。

Automorphism groups of endomorphism semigroups of free periodic groups

In this paper we describe the automorphism groups of the endomorphism semigroups of free Burnside groups $B(m,n)$ for odd exponents $n\ge1003$. We prove, that the groups $Aut(End(B(m,n)))$ and $Aut(B(m,n))$ are isomorphic. In particular, if the groups $Aut(End(B(m,n)))$ and $Aut(End(B(k,n)))$ are isomorphic, then $m=k$.

Centers of Sylow Subgroups and Automorphisms

Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovk…

Coprime automorphisms of finite groups

Let $G$ be a finite group admitting a coprime automorphism $α$ of order $e$. Denote by $I_G(α)$ the set of commutators $g^{-1}g^α$, where $g\in G$, and by $[G,α]$ the subgroup generated by $I_G(α)$. We study the impact of $I_G(α)$ on the structure of $[G,α]$. Suppose that each subgroup generated by a subset of $I_G(α)$ can be generated by at most $r$ elements. We show that the rank of $[G,α]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(α)$ has odd order, then $[G,α]$ has odd order too. Further, i…

奇特征下超奇异 K3 曲面的辛自同构有限群

Finite Groups of Symplectic Automorphisms of Supersingular K3 surfaces in Odd Characteristics

本文利用格理论方法研究奇特征下有限群在 K3 曲面上的辛作用，提出 p-根对机制约束超奇异 K3 曲面的非温顺辛作用，并给出群阶中 p 的指数上界。

Non-inner automorphisms of order p in finite p-groups of coclass 4 and coclass 5

A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner automorphism of order $p$. In this paper we prove the conjecture for finite $p$-groups of coclass $4$ and coclass $5$ ($p\ge 5$). We also prove the conjecture for an odd order nonabelian $p$-group $G$ with cyclic center satisfying $Z_3(G)\cap C_G(G^pγ_3(G))\le Z(Φ(G))$.

A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions

The pro-isomorphic zeta function of a finitely generated nilpotent group $Γ$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $Γ$. Such zeta functions can be expressed as Euler products of $p$-adic integrals over the $p$-adic points of an algebraic automorphism group associated to $Γ$. In this way they are closely related to classical zeta functions of algebraic groups over local fields. We describe the algebraic automorphism groups for a natural family of class-$2$ nilpotent groups; these groups can be viewed as gene…

Minimal generating sets in wreath products

We find some sufficient conditions under which the permutational wreath product of two groups has a minimal generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions.

On some rational extension properties for $GL_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms

In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.

具有循环中心的有限p-群中阶为p的非内自同构

Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

本文研究具有循环中心的有限非交换p-群，证明在中心化子不包含于特定子群$γ_{3}(G)G^{p}$的条件下，该群必存在阶为p的非内自同构。

A note on commuting automorphisms of some finite $p$-groups

An automorphism $α$ of a group $G$ is called a commuting automorphism if each element $x$ in $G$ commutes with its image $α(x)$ under $α$. Let $A(G)$ denote the set of all commuting automorphisms of $G$. Rai [Proc. Japan Acad., Ser. A {\bf 91} (2015), no. 5, 57-60] has given some sufficient conditions on a finite $p$-group $G$ such that $A(G)$ is a subgroup of Aut$(G)$ and, as a consequence, has proved that in a finite $p$-group $G$ of co-class 2, where $p$ is an odd prime, $A(G)$ is a subgroup of Aut$(G)$. We give here very elementary and short proofs of main results of Rai.

Rigid automorphisms of linking systems

A rigid automorphism of a linking system is an automorphism which restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian, and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group $G$ restricts to the identity on the centric lin…

On the Groups and Automorphism Groups of the Groups of order 64p without a Normal Sylow p-Subgroup

The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those groups of order 64 with an odd-order automorphism. In view of the fact that we already had determined these groups and that these automorphism groups are not given explicitly in the literature, we have appended to this report these automorphism groups. In another project we were looking for new complete groups by following automorphism group towers up to comple…

On minimal finite quotients of outer automorphism groups of free groups

We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F_n, for n = 3 and 4, using presentations of these groups.

$p$ 为奇素数时 $p^6$ 阶群的忠实置换表示的最小次数

Minimal degrees for faithful permutation representations of groups of order $p^6$ where $p$ is an odd prime

本文确定了奇素数 $p$ 下所有 $p^6$ 阶群的忠实置换表示的最小次数，并给出了构造方法。

Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups

A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

Finite $p$-groups of class 3 with noninner automorphisms of order $p$

A longstanding conjecture asserts that every non-abelian finite $p$-group $G$ admits a non-inner automorphism of order $p$. The conjecture is valid for finite $p$-groups of class 2. Here, we prove every finite non-abelian $p$-group $G$ of class 3 with $p>2$ has a noninner automorphism of order $p$ leaving $Φ(G)$ elementwise fixed. We also prove that if $G$ is a finite 2-group of class 3 which cannot be generated by 4 elements, then $G$ has a non-inner automorphism of order 2 leaving $Φ(G)$ elementwise fixed. We also prove that the latter conclusion holds for finite 2-groups $G$ of class 3 such…

Outer automorphisms of classical algebraic groups

The so-called Tits class, associated to an adjoint absolutely almost simple algebraic group, provides a cohomological obstruction for this group to admit an outer automorphism. If the group has inner type, this obstruction is the only one. In this paper, we prove this is not the case for classical groups of outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$. More precisely, we prove a descent theorem for exponent $2$ and degree $6$ algebras with unitary involution, which shows that their automorphism groups have outer automorphisms. In all other relevant classical t…

关于被某个域自同构固定的不可约特征标次数与p-可解群

On the degrees of irreducible characters fixed by some field automorphism, p-solvable groups

本文将实值不可约特征标奇次数蕴含正规Sylow 2-子群的结论推广到p-可解群，证明若p不整除被p阶域自同构固定的不可约特征标次数，则该群具有正规Sylow p-子群。

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, $p$ …

斜括号的Baer变换

The Baer Transform for Skew Braces

研究奇数阶且幂零类至多为2的有限斜括号的Baer变换。从Baer对奇数阶幂零类2的幂零群的经典构造出发，用阿贝尔群替换加法群，证明得到的是阿贝尔型有限斜括号，还研究了其性质及应用。

关于斜近生的自同构群平凡性与非平凡性的研究

On the triviality and non-triviality of the automorphism group of a skew brace

本文证明了某些斜近生家族的自同构群平凡性与群阶关系，且构造了阶为2p³的斜近生具有平凡自同构群。

具有半二面体Sylow 2-子群的有限群的类保持Coleman自同构群

Class-preserving Coleman Automorphisms of Finite Groups with Semidihedral Sylow 2-Subgroups

本文研究了具有半二面体Sylow 2-子群的有限群的类保持Coleman自同构群，证明其具有奇数阶，并扩展了相关文献中的结论。

关于有限阿贝尔群自同构的若干回答

An answer regarding automorphisms of finite abelian groups

本文否定了一个关于有限阿贝尔群自同构群大小与群大小比值的猜想，证明该比值为有理数时分母必须平方自由，并且奇素数不能作为此类比值。

On finite $p$-groups with abelian automorphism group

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $γ_2(G) < \mathrm{Z}(G) < Φ(G)$, where $γ_2(G)$, $\mathrm{Z}(G)$ and $Φ(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\mathrm{Z}(G) = Φ(G)$ is elementary abelian; (ii) $γ_2(G) = Φ(G)$ is elementar…

Mazurov关于自由Burnside群自同构不动点的问题

Mazurov's question on fixed points of automorphisms of free Burnside groups

本文否定回答了Mazurov关于自由Burnside群自同构不动点的问题，证明对任意奇数m≥3及足够大的素数周期q，循环置换生成元的m阶自同构无不动点。

Subgroups of odd order in the real plane Cremona group

In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.

On the automorphism groups of rank-4 primitive coherent configurations

The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive coherent configuration with $n$ vertices is not a Cameron scheme, then its automorphism group has minimal degree $\geq cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this pap…

作为对称设计的旗传递且点本原自同构群的仿射群

Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs

本文研究了容许旗传递且点本原仿射自同构群的对称设计，证明了当λ为素数时自同构群的结构，并给出了此类对称设计的完整分类。

Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or $G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.

Quandle代数的幂等元、自同构群与换位子宽度

Idempotents, automorphism groups, and commutator widths of quandle algebras

本文研究Quandle代数，证明特征不为2的整环上有序交换Quandle的Quandle代数无非平凡幂等元，计算两类Quandle代数的自同构群，通过搜索得到换位子宽度为2的首批Quandle代数例子。

Permutation groups and derangements of odd prime order

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.

内自同构群的极小性

Minimality of the inner automorphism group

本文提出了稠密子群的$z$-极小性判定准则，并证明了连通Lie群的内自同构群极小性的充要条件，同时探讨了局部紧幂零群与亚交换群的$z$-极小性。

Automorphisms of real del Pezzo surfaces and the real plane Cremona group

We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting minimally on them. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.

有限群中具有缠绕Sylow 2-子群的Coleman自同构

Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups

该研究证明了具有缠绕Sylow 2-子群的有限群的外自同构交集具有奇数阶，并解决了三种特定2-群的正规化问题。

Smallest non-cyclic quotients of the automorphism group of free groups

We give an new, elementary proof of the result that the smallest non-cyclic quotients of automorphism group of free group is the linear group over the field of two elements, and moreover all minimal quotients are obtained by the standard projection composed with an automorphism of the image. This result, originally due to Baumeister-Kielak-Pierro, proves a conjecture of Mecchia-Zimmermann.

All p-local finite groups of rank two for odd prime p

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.

On the Vanishing Criterion for the Cohomology Groups of the Automorphism Group of a finite Abelian $p$-Group

For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite abelian $p$-group $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathbb{Z}/p^{λ_i}\mathbb{Z})^{ρ_i}$, where $p$ is a prime, we consider two actions of its automorphism group $\mathcal{G}_{\underlineλ}$ on $\mathcal{A}_{\underlineλ}$. The first action is the natural action $g\bullet a=\ ^ga$ for all $g\in\mathcal{G}_{\underlineλ}$ and $a\in\mathcal{A}_{\underlineλ}$ where the action map is denoted by $Λ_1=Id_{\mathcal{G}_{\underlineλ}}:\mathcal{G}_{\underlineλ}\lo…

Units of commutative group algebra with involution

Let p be an odd prime, F the field of p elements and G a finite abelian p-group with an arbitrary involutory automorphism. Extend this automorphism to the group algebra FG and consider the unitary and the symmetric normalized units of FG. This paper provides bases and determines the invariants of the two subgroups formed by these units.

On the automorphism group of the m-coloured random graph

Let $R_m$ be the (unique) universal homogeneous $m$-edge-coloured countable complete graph ($m\ge2$), and $G_m$ its group of colour-preserving automorphisms. The group $G_m$ was shown to be simple by John Truss. We examine the automorphism group of $G_m$, and show that it is the group of permutations of $R_m$ which induce permutations on the colours, and hence an extension of $G_m$ by the symmetric group of degree $m$. We show further that the extension splits if and only if $m$ is odd, and in the case where $m$ is even and not divisible by~$8$ we find the smallest supplement for $G_m$ in its …

带算子的域的自同构群的单纯性

Simplicity of the automorphism group of fields with operators

本文改编 Lascar 的证明，证明了涵盖强极小理论和带算子域理论的一类不可数模型中，固定所有非泛型元素的自同构群具有单纯性。

On the Automorphism Group of a Binary Self-dual [120, 60, 24] Code

We prove that an automorphism of order 3 of a putative binary self-dual [120, 60, 24] code C has no fixed points. Moreover, the order of the automorphism group of C divides 2^a.3.5.7.19.23.29 where a is a nonegative integer. Automorphisms of odd composite order r may occur only for r=15, 57 or r=115 with corresponding cycle structures 15-(0,0,8;0), 57-(2,0,2;0) or 115-(1,0,1;0), respectively. In case that all involutions act fixed point freely we have |Aut(C)|<=920, and Aut(C) is solvable if it contains an element of prime order p>=7. Moreover, the alternating group A_5 is the only non-abelian…