A complete description of a, pgroups is given by means of a, ptypes. Aknowledgement the authors would like to thank the referee for his remarks and corrections. Pdf automorphisms of finite abelian groups researchgate. Mar 17, 2014 such groups are then described into some details by considering automorphisms of abelian groups. Let g be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree d in the complex projective space p. Then g is isomorphic to a product of groups of the form hp zpez x x zpez. Pdf the automorphism group of a free group is complete. Torsionfree abelian groups with finite groups of automorphisms v. Abelian groups of rank 0 are precisely the periodic groups, while torsionfree abelian groups of rank 1 are necessarily subgroups of and can be completely described. Finally, it is shown that if g is a locally graded group with the property that the above index is. Normal automorphisms of a soluble product of abelian groups. Theorem 2 identifies inertial automorphisms of torsionfree abelian groups, while theorem 3 treats. Outer automorphism groups are of general interest to many geometric group theorists, with a signi cant amount of attention paid to mapping class groups outer automorphism groups of closed surface groups, outf n, and there is growing interest in outer automorphism groups of rightangled artin groups.
Finite groups of automorphisms of infinite groups, i sciencedirect. More generally, a torsionfree abelian group of finite rank r \displaystyle r is a. The cardinality of s is called the rank of the free group. Suppose more that there exists an element g in g with order a primepower, such that g. In category theory, an automorphism is an endomorphism i. The author introduces for a, p groups the concept of a, ptype, which represents a choice of k integer matrices. We prove that the subgroup of all automorphisms of g identical on the. There is no condition that ensures that a group has abelian aut g. Automorphism groups of free groups, surface groups and free. Some new nonabelian 2groups with abelian automorphism groups by alireza jamali, journal of group theory, issn 14435883 print, issn 14434446 online, volume 5,number 1, page 53 57year 2002. We do not concern ourselves with the apparently hopeless task of finding all the torsionfree abelian groups whose automorphism group is a given finite group. It is relatively easy to see that finitely generated abelian groups do not qualify, so if such a group exists, it has to be infinitely generated.
A propgroup is a projective limit of finite pgroups, or equivalently a compact. Normal forms for automorphisms of universal coxeter groups. Automorphisms of free groups with a free group being one of the most basic examples of an in nite group, the group of outer automorphisms of a free group f. Automorphism groups of free groups, surface groups and free abelian groups martin r.
Normal automorphisms of free profinite groups sciencedirect. It is known that in abelian groups 4 and in the absolutely free groups 6, every weak automorphism is a superposition of an automorphism with 1 or simply an automorphism. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. In other words, we give another construction of moritas 1cocyle with the sl2, crepresentations of the free abelian group.
The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. This is not the first time that the theory of abelian groups has been used in understanding the automorphisms of nonabelian pgroups. Automorphism groups of free groups, surface groups and. Using this description, a series of problems on finite groups of automorphisms of torsionfree abelian groups of. In this paper we study the images of the johnson homomorphisms of the basisconjugating automorphism groups of free groups and free metabelian groups. A complete description of a, p groups is given by means of a, ptypes. We show that our crossed homomorphism induces moritas 1cocycle defined in 22. Under composition, the set of automorphisms of a graph forms what algbraists call a group. Torsionfree abelian groups with finite groups of automorphisms. Finally, an isomorphism has an inverse which is an isomorphism, so the inverse of an automorphism of gexists and is an automorphism of g. A group is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if its automorphism group is an abelian group or equivalently, if any two automorphisms of the group commute. What are the automorphism groups of principally polarized. Automorphism group of finitely generated free group groupprops.
Since ig is an invertible homomorphism, its an automorphism. In the literature there are a few examples of non abelian p groups with commutative autg. This reduction is carried out by appealing to a fact about product automorphisms for groups with relatively prime numbers of elements lemma 2. A propgroup is a projective limit of finite p groups, or equivalently a compact. Fs is the image of s under some automorphism of fs, then s is. In addition we prove that the group of unitriangular automorphisms of a free associative. The automorphism group of a free group of finite rank has been investigated inten sively. There is no condition that ensures that a group has abelian autg. Journal of algebra 80, 106112 1983 finite groups of automorphisms of infinite groups thomas a. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.
Notation we follow the notation of 12 and 10, where basic results about propgroups can be found. In introductory abstract algebra classes, one typically en counters the classification of finite abelian groups 1. Such groups are then described into some details by considering automorphisms of abelian groups. Finite groups of automorphisms of infinite groups ii. This volume features seventeen extended conference abstracts corresponding to selected talks given by participants at the crm research program automorphisms of free groups. In mathematics, a free abelian group or free zmodule is an abelian group with a basis, or, equivalently, a free module over the integers. Notation we follow the notation of 12 and 10, where basic results about prop groups can be found.
In the literature there are a few examples of nonabelian pgroups with commutative autg. Fournelle department of mathematics, university of alabama, university, alabama 35486 communicated by marshall hall, jr. Abelian automorphisms groups of schottky type 191 index normal subgroup. For a given set a, let us agree to say that a bijection f. On the basisconjugating automorphism groups of free groups. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. Abelian automorphisms groups with a finite number of orbits. Let be a full subcategory of the category of abelian.
On the automorphisms of a graph product of abelian groups. There are some papers in which the authors are looking for group operations other than x y or y x, which are not induced by a weak automorphism. Each section is followed by a series of problems, partly to check understanding marked with the letter \r. We will study the group outf, of outer automor phisms of f, by studying its action on a space x, which is analogous to the. Weak automorphisms of dihedral groups pdf paperity. Extended abstracts fall 2012 automorphisms of free groups. Automorphisms of infinite abelian groups mathematics stack. Bridson and karen vogtmann 23 july 2005 the group of 2. That is, in an abelian group the inner automorphisms are trivial.
It turns out that this automorphism is surjective in other words, every automorphism of the free abelian. Recommended problem, partly to present further examples or to extend theory. The author introduces for a, pgroups the concept of a, ptype, which represents a choice of k integer matrices. Automorphisms of finite groups inder bir singh passi springer. Gare also characterized by the lifting property in the category of groups. Our results have a number of applications which are geometric in nature. Automorphisms of abelian group extensions article pdf available in journal of algebra 3244 november 2009 with 190 reads how we measure reads. We nd our lower bound by constructing a free abelian subgroup of outa. The free abelian group of rank is isomorphic to, and its automorphism group is isomorphic to. In 24, nielsen obtained its presentation in terms of. This is a very abstract definition since, in category theory, morphisms arent necessarily functions and objects arent necessarily sets. Pdf on abelian automorphism groups of hypersurfaces. Gr 29 jul 2005 automorphism groups of free groups, surface groups and free abelian groups martin r. Note that for topological automorphisms of topological groups fixed points are known.
I have more notes on groups whose automorphism group is abelian here and here. We now introduce the groups which are studied in the present article. This is not the first time that the theory of abelian groups has been used in understanding the automorphisms of non abelian p groups. Automorphisms of finite abelian groups article pdf available in the american mathematical monthly 11410 june 2006 with 256 reads how we measure reads. On the sl2, crepresentation rings of free abelian groups. For the upper bound, we reduce the problem to that of nding the vcd of the subgroup p n. The book elucidates automorphisms of groups is a fundamental topic of study in group theory and explores an exposition of various developments on some wellknown problems regarding the relationship between orders of finite groups and that of their automorphism groups. A a has the autoproperty if it is possible to nd a binary operation.
Received october 7, 1980 this paper studies aut a in the case in which a is an infinite abelian group and aut. Ushakov 1 mathematical notes volume 70, pages 830 837 2001 cite this article. We will show that this matrix is a product of the matrices of the form 3. Group whose automorphism group is abelian groupprops. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and provethat everyelement from the derived subgroupis acommutator. A basis is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with. Automorphisms abstract an automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Torsionfree groups having finite automorphism groups. If n k, this is known as the pure symmetric group and was shown by collins. Nikiforov 1 mathematical notes of the academy of sciences of the ussr volume 39, pages 350 353 1986 cite this article.
Automorphism groups of free groups, surface groups and free abelian groups. In general very little is known about groups with abelian automorphisms. In 1, we characterized the automorphisms of abelian pgroups having the extension property in the category of abelian pgroups, as well as those having the extension property in the category of all abelian groups. Using this description, a series of problems on finite groups of automorphisms of torsion free abelian groups of. It is wellknown that apart from cyclic groups no commutative group has abelian automorphism group. Automorphisms of finite abelian groups christopher j. Let g be an abelian group and a an abelian group of automorphisms of g such that g is a union of a. The set of inner automorphisms of gis denoted inng. On automorphisms of free centerbymetabelian and nilpotent. Free abelian groups are free algebras in the variety of abelian groups. By work of day day09, any automorphism of a decomposes uniquely into an f. On group whose inner automorphisms groups of order 6. The material on free groups, free products, and presentations of groups in terms of generators and relations see earlier handout on describing. Abelian groups of rank 0 are precisely the periodic groups, while torsion free abelian groups of rank 1 are necessarily subgroups of and can be completely described.
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