It is well known that 1 is the only normal psubgroup of a g if g is elementary. This is because automorphisms are permutations on the underlying set. Faudree, a note on the automorphism group of a p group. If g is not abelian then the centre is a proper normal subgroup and. The second part offers an account of important developments on a conjecture that a finite group has at least a prescribed number of automorphisms if the order of the group is sufficiently large. Automorphisms of finite abelian groups christopher j. Pdf on the automorphisms of a graph product of abelian. Notice, however, that there is a largest possible class for each order. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. In general very little is known about groups with abelian automorphisms. Automorphisms an automorphism of a design is an isomorphism of a design with itself. In the literature there are a few examples of non abelian p groups with commutative autg. A question on automorphisms of finite abelian groups.
For finite groups, being abelian and the automorphism group being abelian as well implies cyclic. As an application, we discuss the notion of natural states in quantum abelian chernsimons theory. The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group g, denoted by autg, forms itself a group, the automorphism group of g. For the proof of the following well known result, one can see 26, pg 69. Our results have a number of applications which are geometric in nature.
That is, in an abelian group the inner automorphisms are trivial. That is, we claim that v is a direct sum of simultaneous eigenspaces for all operators in g. Then g is isomorphic to a product of groups of the form hp zpez x x zpez. Automorphisms of the symmetric and alternating groups. It is shown that every noncentral normal subgroup of t contains a noncentral elementary abelian normal psub group of t of rank at least 2. Subgroups of finite abelian groups and automorphisms. In particular, g always has nilpotency class 2 when 0. Is there a non cyclic group with a cyclic automorphism group. There is no condition that ensures that a group has abelian autg. Examples of noncyclic group with a cyclic automorphism group.
Let t be the automorphism group of a nonelementary reduced abelian pgroup, p 5. This reduction is carried out by appealing to a fact about product automorphisms for groups with relatively prime numbers of elements lemma 2. Automorphisms of abelian groups sage reference manual v9. Formalisms in terms of the supergroup property collapse operator this group property can be defined in terms of the collapse of. When n 6, the symmetric group has a unique nontrivial class of outer automorphisms, and when n 2, the symmetric group, despite having no outer automorphisms, is abelian, giving a nontrivial center, disqualifying it from being complete. Informally, it is a permutation of the group elements such that the structure remains unchanged. A a has the autoproperty if it is possible to nd a binary operation. On automorphisms of finite abelian p groups springerlink. Central automorphisms let aut be the group of central automorphisms of a group cg g.
For a given set a, let us agree to say that a bijection f. Showing that a cyclic automorphism group makes a finite. Introduction in introductory abstract algebra classes, one typically encounters the classi. Then for any abelian group g, theorem 2i follows by the fact that a transitive and abelian permutation group. In all other cases 0 is either trivial or the direct product of a finite number of groups of order two. The fact that acgjag is not trivial can either be viewed as a special instance where g admits noninner automorphisms which are central and classpreserving, or be viewed as a special instance where automorphisms of finite groups 59 hr, m 0, r is the group of units acting by multiplication on the additive group of the ring. A group is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if is an abelian group. A group automorphism is a group isomorphism from a group to itself.
In introductory abstract algebra classes, one typically en counters the classification of finite abelian groups 1. We study inertial automorphisms of abelian groups and give characterization of them. For all abelian groups there is at least the automorphism that replaces the group elements by their inverses. Pdf automorphisms of finite abelian groups researchgate. Automorphism groups of free groups, surface groups and.
Finite pgroups with the least number of outer pautomorphisms. Any group whose automorphism group is abelian must have nilpotency class at most two, because the inner automorphism group, being a subgroup of the automorphism group, is abelian. When v is onedimensional over the field zpz of p elements, g is the unique non abelian group of order p 3 and exponent p the group in question. Graph automorphisms department of electrical engineering. If gis not abelian then the centre is a proper normal subgroup and. For brevity we shall say that a group g is sa if every semiautomorphism of g is an automorphism or an antiautomorphism. In the infinite case, there are locally cyclic groups that are not. A remark about central automorphisms of groups giovanni cutolo to guido zappa on his 90th birthday the aim of this paper is to prove and discuss the following simple result on central automorphisms, that, although elementary and not surprising at all, and therefore possibly known, seems to have not been recorded in the literature yet. Abelian extensions, automorphisms of groups, cohomology of groups, split. Abelian extensions, automorphisms of groups, cohomology of groups, split extensions. Abelian group 1 abelian group in abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order the axiom of commutativity. Since autcg inng for all abelian groups g, it is su. Inng is a normal subgroup of the full automorphism group autg of g.
In this article, we present an extensive survey on the developments in the theory of non abelian finite groups with abelian automorphism groups, and pose some problems and further research directions. In 1928 shoda 10 described the structure of the automorphism group of a finite abelian group. With the addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a. Bridson and karen vogtmann 23 july 2005 the group of 2. In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of s 6, the symmetric group on 6 elements. It is shown that every noncentral normal subgroup of t contains a noncentral elementary abelian normal psubgroup of t of rank at least 2. Fournelle department of mathematics, university of alabama, university, alabama 35486 communicated by marshall hall, jr. For which abelian groups g does the group of automorphisms of ggenerate. Harriss 1980 work on p automorphisms of abelian p groups.
Automorphisms of finite abelian groups article pdf available in the american mathematical monthly 11410 june 2006 with 253 reads how we measure reads. The automorphism group of a design is always a subgroup. Automorphisms of finite abelian groups 3 as a simple example, take n 3 with e1 1, e2 2, and e3 5. Then gis isomorphic to a product of groups of the form h p zpe1z. In other words, it gets a group structure as a subgroup of the group of all permutations of the group. On automorphisms of cayleydigraphs of abelian groups. On the automorphisms of a graph product of abelian groups. The smallest nonsolvable group is the simple group a 5, the alternating group of order 60 inside. Under composition, the set of automorphisms of a graph forms what algbraists call a group.
Thus there is an easy characterization of pgroups of class 1. Group whose automorphism group is abelian groupprops. A non abelian group of primepower order is said to have divisibility property if its order divides that of its automorphism group. Automorphism groups of free groups, surface groups and free. The main tools on the proof is an extension of the methods in sw15. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules and vector spaces, are developed. Representation theory of nite abelian groups october 4, 2014 1.
The theory of abelian groups is generally simpler than that of their non abelian counterparts, and finite abelian groups are very well understood. An isomorphism from x to itself is called an automorphism of x. The overflow blog how the pandemic changed traffic trends from 400m visitors across 172 stack. Throughout the following, g is a reduced pprimary abelian group, p 5, and v is the group of all automorphisms of g. Those sub groups of primary abelian groups which are complete groups of fixed elements. Every nite group g has a composition series, and the collection of successive quotients is a collection of nite simple groups independent of the choice of composition series.
Pdf automorphisms of abelian group extensions researchgate. B, g arbitrary nite group, by a result of chow 3, which asserts that if all eigenvalues of a multidigraph are simple, then its automorphism group is abelian. It turns out that matrix multiplication also makes this set into a ring as. Journal of algebra 80, 106112 1983 finite groups of automorphisms of infinite groups thomas a. A survey 4 in this part we consider p groups of small orders and mention results on classpreservingautomorphismsof pgroupsoforderat most p5. The automorphism group of a design is always a subgroup of the symmetric group on v letters where v is the number of points of the design. We denote the set of all automorphisms on x as autpxq. Primary abelian groups and their automorphisms jstor. If the inner automorphism group of a perfect group g is simple, then g is called quasisimple. Earnley, on finite groups whose group of automorphisms is abelian, ph. Received october 7, 1980 this paper studies aut a in the case in which a is an infinite abelian group and aut. Automorphisms of abelian group extensions article pdf available in journal of algebra 3244 november 2009 with 186 reads how we measure reads. 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.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. A generated by the inertial automorphisms of an abelian group a, that is, automorphisms. We study the automorphisms of a graph product of finitelygenerated abelian groups w. Sums of automorphisms of a primary abelian group mathematical. I deleted my answer i think, and instead typed everything up into a pdf. We say x and y are isomorphic and write x y if there exists a bijection vpxqnvpyqsuch that uv pepxqif and only if. Automorphism groups of abelian p groups jutta hausen1 abstract. The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. Automorphism groups of free groups, surface groups and free abelian groups martin r. When v is onedimensional over the field zpz of p elements, g is the unique nonabelian group of order p 3 and exponent p the group in question. Prescribing symmetries and automorphisms for polytopes. Browse other questions tagged abstractalgebra group theory abelian groups automorphism group or ask your own question. Finite groups of automorphisms of infinite groups ii.
View enhanced pdf access article on wiley online library html view download pdf for offline viewing. For every group g there is a natural group homomorphism g autg whose image is the group inng of inner automorphisms and whose kernel is the center of g. This research was partially supported by national science foundation grant mcs7606001. Automorphisms abstract an automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges.
The automorphisms of primary abelian groups hill 1971. Jun 15, 2008 let a be a finitely generated abelian group. Pdf automorphisms of abelian group extensions manoj. The fact that we are considering automorphisms g ginstead of the more general homomorphisms g his solely.
The set of all automorphisms of a design form a group called the automorphism group of the design, usually denoted by autname of design. However, most pgroups are of class 2, in the sense that as n. Sanders 16 used a technique quite similar to ours to count the number of central automorphisms in a finite pn purely non abelian p group. It is wellknown that apart from cyclic groups no commutative group has abelian automorphism group. The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of g is a group, the inner automorphism group of g denoted inng. Let a be the group of all automorphisms of the finite. Browse other questions tagged group theory finite groups abelian groups or ask your own question. More precisely, we study a natural subgroup aut w of aut w, with aut w aut w whenever vertex groups are. The material on free groups, free products, and presentations of groups in terms of generators and relations see earlier handout on describing. The solvable groups are thus those groups whose simple successive quotients in a composition series are prime cyclic abelian groups. We describe the automorphism group auta using the rank of a and its torsion part ppart a p. The purpose of this paper is to study the problem of extending automorphisms of.