Marco Ruscitti, Leire Legarreta, Manoj K. Yadav
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.
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Automorphism groups of $2$-groups of coclass at most $3$ are investigated.
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 $Z_3(G)\cap C_G(G^pγ_3(G))\le Z(Φ(G))$.
Every finite $p$-group of coclass 2 has a noninner automorphism of order $p$ leaving the center elementwise fixed.
It is shown that if G is a finite p-group of coclass 2 with p > 2, then G has a noninner automorphism of order p.
In this note, the existence of noninner automorphisms of order 2 for finite 2-groups of coclass 2 is proved. Combining our result with a recent one due to Y. Guerboussa and M. Reguiat (see arXiv:1301.0085), we prove that every finite $p$-group of coclass 2 has a noninner automorphism of order $p$ leaving the center elementwise fixed.
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 $5$ ($p\ge 5$).
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].
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.
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.
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…
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…
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。
In this survey article, we try to summarize the known results towards the long-standing non-inner automorphism conjecture, which states that every finite non-abelian $p$-group has a non-inner automorphism of order $p$.
A Note on Fuzzy Automorphism and Inner Automorphism of Groups
该文将模糊自同态概念拓展至模糊自同构，引入模糊内自同构及模糊子群诱导的模糊内自同构概念，得到经典群论著名结果的模糊对应版本。
In this paper we study the existence of at least one non-inner automorphism of order p of a finite thin p-group, for any prime p.
Some results of cohomological properties of $p$-group and non-inner automorphism with order $p$ on non-abelian finite $p$-group
本文研究了有限 $p$-群的上同调性质，证明了非交换有限 $p$-群存在阶为 $p$ 的非内自同构。
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.
Characterizing inner automorphisms and realizing outer automorphisms
本文用纯群论方法给出有限群自同构的两个定理的初等证明：内自同构的扩张刻画，以及任意有限群可实现为某有限群的外自同构群。
Let G be a finite p-group and let Aut_l(G) be the group of absolute central automorphisms of G. We give necessary and sufficient conditions on G such that Aut_l(G) = Inn(G).
Let $G$ be a finite $p$-group.
An improved characterisation of inner automorphisms of groups
本文证明任意群G可嵌入到一个简单、完备的co-Hopfian群H中，非平凡自同态仅在为内自同构时可延伸到H，改进了Schupp的定理。
On the non-inner automorphism conjecture of finite $p$-groups
针对有限非交换 $p$-群必含阶为 $p$ 的非内自同构这一长期猜想，本文证明其在满足特定条件、幂零类为 $n$ 且 $p>2$ 的有限 $p$-群中成立。
On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor
本文证明了带有非平凡阿贝尔直因子的有限非阿贝尔 $p$-群存在阶为 $p$ 的中心非内自同构，且该自同构逐点固定 $Φ(G)$，从而验证了长期存在的猜想。
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 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…
The Inverse Monoid of Partial Inner Automorphisms of a Semigroup
研究半群的偏内自同构逆单群结构，分析其在群、完全简单半群、全变换单群及有限Abel群作用下的有限G集的内单群中的表现，并提出开放问题。
On the conjecture of non-inner automorphisms of finite $p$-groups
针对有限非交换 $p$-群必有阶 $p$ 非内自同构的猜想，本文在奇数阶单块群且极大子群满足中心换位条件时，构造并证明了逐元固定 $\Phi(G)$ 的阶 $p$ 非内自同构。
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…
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).
We study class preserving automorphisms of groups of order thirty two and prove that only two groups have non-inner class preserving automorphisms.
In this paper we classify all finite 2-groups of class 2 for which every automorphism of order 2 leaving the Frattini subgroup elementwise fixed is inner. We prove that every such group G is isomorphic to Q(n; r) = <a, b| a^{2n}= b^{2r}= 1; a^2^{n-r}= [a, b]> for some positive integers r; n such that 2 < 2r <= n; and every automorphism of Q(n; r) of order 2 leaving the Frattini subgroup elementwise fixed is inner.
The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.
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…
The description of the automorphism group of group $<a, b; [a^m,b^n]=1>$ ($m,n>1$) in terms of generators and defining relations is given. This result is applied to prove that any normal automorphism of every such group is inner.
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…
Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of "inner automorphism" applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation…
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。
An automorphism of a group is called outer if it is not an inner automorphism. Let $G$ be a finite $p$-group. Then for every outer $p$-automorphism $ϕ$ of $G$ the subgroup $C_G(ϕ)=\{x\in G \;|\; x^ϕ=x\}$ has order $p$ if and only if $G$ is of order at most $p^2$.
Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $α$ an automorphism of $G$ that satisfies $α^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\mathrm{Dih}(m,G,α)$ is defined on $\mathbb Z_m\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)α^{ij})$. We prove that two finite dihedral-like automorphic loops $\mathrm{Dih}(m,G,α)$, $\mathrm{Dih}(\overline{m},\overline{G},\overlineα)$ are isomorphic if and only if $m=\overline{m}$, $G=\overline{G}$, and…
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.
We discuss the following question of G. Makanin from ``Kourovka notebook'': does there exist an algorithm to determine is for an arbitrary pair of words $U$ and $V$ of a free group $F_n$ and an arbitrary automorphism $ϕ\in Aut(F_n)$ the equation $ϕ(X) U = V X$ solvable in $F_n$? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of $F_n$.
We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the description of involutions in automorphism groups of free groups (J. Dyer, P. Scott, 1975) we obtain a group-theoretic characterization of inner automorphisms determined by primitive elements in the automorphism group of any non-abelian free group F. It follows that the subgroup Inn(F) is characteristic in Aut(F), and hence the latter one is complete.
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results.
On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups
研究acylindrically hyperbolic群G的自同构群对其同调拟同构空间的作用，确定其核为强共轭化自同构子群，并证明若G无非平凡有限正规子群，则其有足够多的拟同构以识别自同构是否内蕴。
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…
We study properties of automorphisms of graph products of groups. We show that graph product $Γ\mathcal{G}$ has non-trivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has non-trivial pointwise inner automorphisms. We use this result to study residual finiteness of $\mathop{Out}(Γ\mathcal{G})$. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then $\mathop{Out}(Γ\mathcal{G})$ is residually finite. Finally, we g…
Let $G$ be a finite $p$-group of order $p^5$, where $p$ is a prime. We give necessary and sufficient conditions on $G$ such that $G$ has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of section 5 of Yadav [Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008), 1-11] and Theorem 4.2 of Kalra and Gumber [Indian J. Pure Appl. Math. 44 (2013), 711-725].
Homogeneous quandles with abelian inner automorphism groups
本文刻画了具有阿贝尔内自同构群的齐次拟群，证明其可表示为平凡拟群的阿贝尔扩张，并推广了Furuki和Tamaru构造不连通平坦拟群的工作。
An automorphism $α$ of a group $G$ is said to be central if $α$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter examples to a conjecture of A. Mahalanobis [Israel J. Math., {\bf 165} (2008), 161 - 187]. We also construct a family of finite $p$-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in group theory, Aracne Editrice, Rome 2002, 111-127].
We give a sufficient condition on a finite $p$-group $G$ of nilpotency class 2 so that $\Aut_c(G) = \Inn(G)$, where $\Aut_c(G)$ and $\Inn(G)$ denote the group of all class preserving automorphisms and inner automorphisms of $G$ respectively. Next we prove that if $G$ and $H$ are two isoclinic finite groups (in the sense of P. Hall), then $\Aut_c(G) \cong \Aut_c(H)$. Finally we study class preserving automorphisms of groups of order $p^5$ and prove that $\Aut_c(G) = \Inn(G)$ for all the groups of order $p^5$ except two isoclinism families.
Any group $G$ gives rise to a 2-group of inner automorphisms, $\mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $\mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $\mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $\mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(…
We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
We prove that even Coxeter groups, whose Coxeter diagrams contain no (4,4,2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.
In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.
To each totally disconnected, locally compact topological group G and each group A of automorphisms of G, a pseudo-metric space of ``directions'' has been associated by U. Baumgartner and the second author. Given a Lie group G over a local field, it is a natural idea to try to define a map from the space of directions of analytic automorphisms of G to the space of directions of automorphisms of the Lie algebra L(G) of G, which takes the direction of an analytic automorphism of G to the direction of the associated Lie algebra automorphism. We show that, in general, this map is not well-defined.…
We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).
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…
Let $G$ be a finite group and $M,N$ be two normal subgroups of $G$. Let $Aut_N^M(G)$ denote the group of all automorphisms of $G$ which fix $N$ element wise and act trivially on $G/M$. Let $n$ be a positive integer. In this article we have shown that if $G$ and $H$ are two $n$-isoclinic groups, then there exists an isomorphism from $Aut_{Z_n(G)}^{γ_{n+1}(G)}(G)$ to $Aut_{Z_n(H)}^{γ_{n+1}(H)}(H)$, which maps the group of $n^{th}$ class preserving automorphisms of $G$ to the group of $n^{th}$ class preserving automorphisms of $H$. Also, for a nilpotent group of class at most $(n+1)$, with some s…
Let $G$ be a group. The central automorphism group $Aut_c(G)$ of $G$ is the centralizer of $Inn(G)$ the subgroup of $Aut(G)$ of inner automorphisms. There is a one to one map $ σ\mapsto h_σ$ from the set $Aut_c(G)$ onto the set $Hom(G,Z(G))$ of homomorphisms from $G$ onto its center, with $ h_σ(x)=x^{-1} σ(x)$. This map can be used to obtain informations about the size of $Aut_c(G)$, and also about its structure in some special cases. In this paper we see how to use it to obtain informations about the structure of $Aut_c(G)$ in the general case. The notion of the adjoint group of a ring is the…
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