If g is a nonabelian p group, then g has a noninner p automorphism. Complex in a group, product of complexes and related theorems. Endomorphisms and automorphisms a pgroup is homocyclic of exponent eif it is the direct sum of cyclic subgroups all of order pe. In particular, the periodic elements of g form a finite subgroup t, which is a finite p group if g is infinite, and still in this case gt is a torsionfree abelian group whose automorphism group is finite and which therefore has finite quotients modulo charac teristic subgroups of arbitrarily high exponent. A minimal nonabelian group is a nonabelian group such that all its proper subgroups are abelian.
Splitting the automorphism group of an abelian p group. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group. On automorphisms of finite abelian p groups springerlink. The automorphism group of an abelian p group 163 in 5 we have shown that for every noncentral normal subgroup t of ag such that agcr is finite, p 1 divides the index ag. Class equation, p group, definition and related theorems. In mathematical group theory, the automorphism group of a free group is a discrete group of automorphisms of a free group. It is shown that every noncentral normal subgroup of t contains a noncentral elementary abelian normal p sub group of t of rank at least 2. Every inner automorphism is a class automorphism every class automorphism is a family automorphism and a quotientable automorphism. This note considers connected, vertextransitive graphs xof order at least 3, such that the automorphism group of xcontains a.
It is reasonable to conjecture that such a bound exists, that is to say, that a finite group has a prescribed number of automorphisms at least, if only its order is sufficiently large. I want to cite an earlier result that says a homomorphism out of a cyclic group is determined by sending a generator somewhere. Jun 15, 2008 let a be a finitely generated abelian group. A group whose automorphism group is cyclic mathoverflow. G denote the group of all automorphisms of g centralizing g. On the order of the automorphism group of a finite group. If g is a p group then the centre of g is a nontrivial subgroup of g. Automorphism group definition of automorphism group by the. Automorphism groups of algebraic curves with prank zero. A note on the automorphism group of a p group, proc. Nielsen 1924 showed that the automorphisms defined by the elementary nielsen. Cyclic automorphism group implies abelian groupprops.
Finite and cyclic automorphism group implies cyclic. The automorphism group of the quaternions h as a ring are the inner automorphisms, by the skolemnoether theorem. The automorphism group of finite abelian pgroups adolf mader. Cyclic groups september 17, 2010 theorem 1 let gbe an in nite cyclic group. Automorphism groups with cyclic commutator subgroup and. In z p for a prime number p, one nonidentity element can be replaced by any other, with corresponding changes in the other elements. A weaker version of this result was announced by martin in 69, and helleloid and martin 39 prove the general result. In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. In 2 we proved that the full automorphism group of a group is quite rarely isomorphic to a pgroup of maximal class, wherep is a primenever,for instance, if p3. Finite pgroups with a frobenius group of automorphisms whose kernel is a cyclic p group e. That is, for each element g of a p group g, there exists a nonnegative integer n such that the product of p n copies of g, and not fewer, is equal to the identity element. It is conjectured that if g is a flnite noncyclic pgroup of order greater than p2, then jgj divides jautgj.
Automorphism group synonyms, automorphism group pronunciation, automorphism group translation, english dictionary definition of automorphism group. Gis isomorphic to z, and in fact there are two such isomorphisms. If g is a nonabelian pgroup, then g has a noninner pautomorphism. Classification of finite groups with cyclic automorphism group.
On a question about automorphisms of finite pgroups. The automorphism group of an abelian p group and its normal psubgroups by jutta hausen. In particular, for a prime, the automorphism group of the cyclic group of order is the cyclic group of order. Conjugate subgroup, definition and related theorems. Automorphism groups of abelian p groups jutta hausen1 abstract. In mathematics, specifically group theory, given a prime number p, a p group is a group in which the order of every element is a power of p. Loonstra, indagationes mathematicae, volume 74, 1971, 163169.
A presentation of these groups is given in 2, x1, exercise 8a. In mathematics, the automorphism group of an object x is the group consisting of automorphisms of x. Automorphisms of pgroups with cyclic commutator subgroup federico menegazzo. The rank of a homocyclic pgroup h, denoted rkh, is the. We recall the construction and basic properties of finite fields. What is the automorphism group of the additive group of. We prove that the multiplicative group of a finite field is cyclic, and that the automorphism group of a finite field is cyclic generated by the frobenius map. Finite pgroups with a frobenius group of automorphisms. The presentation of the proof in this thesis contains some omitted proof details and revised exposition. This paper deals with the determination of the automorphism group of the metacyclic pgroups, pp, m. In this paper we characterize the flnite nonabelian pgroups g with cyclic frattini. This group is isomorphic to so3, the group of rotations in 3dimensional space. Let t be the automorphism group of a nonelementary reduced abelian p group, p s 5. Automorphisms of pgroup given as cyclicbyelementary abelian central extensions.
Automorphism groups, isomorphism, reconstruction chapter. By definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism. To every nite pgroup one can associate a lie ring lg, and if gg0is elementary abelian then lg is actually a lie algebra over the nite eld gfp. The rst complete characterization of the automorphism group of an abelian group was, however, given by ranum 16. The quotient by the inner automorphisms is the outer automorphism group of a free group, which is similar in some ways to the mapping class group of a surface presentation. It is shown that every noncentral normal subgroup of t contains a noncentral normal subgroup 4 of t such that ap 1. Then, is a cyclic group and hence, a finite cyclic group. The automorphism group of the complete graph kn and the empty graph kn is the symmetric group sn, and these are the only graphs with doubly transitive automorphism groups. For that group all permutations of the three nonidentity elements are automorphisms, so the automorphism group is isomorphic to s 3 and dih 3. Commuting automorphism of pgroups with cyclic maximal subgroups. Autg for all noncyclic pgroups g of order at least p3. Pdf finite groups with abelian automorphism groups.
The purpose of this paper is to apply a similar approach to study the automorphism group of the semidirect product g h k of two cyclic pgroups h and k. G denotes a reduced abelian p group for some prime p. Number of automorphisms of a direct product of two cyclic. Since g is a cyclic p group and p is odd, the automorphism group of g is cyclic 14, 5. Finite p groups with a frobenius group of automorphisms whose kernel is a cyclic p group e. Automorphisms of metacyclic pgroups with cyclic maximal. Journal of algebra 4, 426432 1966 the automorphism group of finite pgroups hans liebeck university of keele, keete, staffordshire, england communicated by p. Finite pgroups with a frobenius group of automorphisms whose. First, assume that g is a finitely presentable group, and n is a cyclic, central subgroup of g, generated. As far as where to go from it, this is roughly how automorphism groups of pgroups are calcuated in obrien 1992, eickleedhamgreenobrien 2002 and the autpgrp package of gap. We describe the automorphism group auta using the rank of a and its torsion part ppart a p.
Since g has odd order, this means that a is the only element of g that is centralized by a. For example, if g is the direct product of two cyclic groups of order 4, generated by a and b, say, and if b is the group of automorphisms consisting of the identity. Once the group has been factored into additive groups, the groups are. The particular interest in the existence of a p group with abelian automorphism group is motivated by a cryptographic protocol proposed in 10. Pdf on the center of automorphism group of a group mohammad. The group g of all inner automor phisms is isomorphic to gz, where z is the centre. 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. G is a normal subgroup of autg the group of all automorphisms and the quotient is called the outer automorphism group of g. In this paper, we characterize the finite pgroups g with cyclic frattini subgroup for which 1aut.
Motivational example the protocol that mahalanobis proposes in article 10 uses nite nonabelian groups based on the di ehellman problem. The class automorphisms form a subgroup of the automorphism group. Journal of algebra 4, 426432 1966 the automorphism group of finite p groups hans liebeck university of keele, keete, staffordshire, england communicated by p. Finite pgroups with the least number of outer pautomorphisms. Every cyclic group of prime order is a simple group which cannot be broken down into smaller groups. A related question concerns the automorphism tower of g, namely. I for a group g, an automorphism of g is a function f. For example, if x is a finitedimensional vector space, then the automorphism group of x is the general linear group of x, the group of invertible linear transformations from x to itself especially in geometric contexts, an automorphism group is also called a symmetry group. Jul 01, 2008 read on the automorphism group of a finite p group with cyclic frattini subgroup, mathematical proceedings of the royal irish academy on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips.
A nice set of generators for the automorphism group of a finite abelian group is described by garrett birkhoff in his paper titled subgroups of abelian groups. To every nite p group one can associate a lie ring lg, and if gg0is elementary abelian then lg is actually a lie algebra over the nite eld gfp. In 2 we proved that the full automorphism group of a group is quite rarely isomorphic to a p group of maximal class, wherep is a primenever,for instance, if p3. Recently it has been proven that a finite pgroup of almost maximal class is an group.
I the inner automorphism group of g, written inng, is the group of automorphisms of the form f gx. This action allows us to completely compute the automorphism group and to derive several of its properties. The automorphism group of a finite metacyclic p group richard m. The automorphism group of the octonions o is the exceptional lie group. Furthermore, for every positive integer n, nz is the unique subgroup of z of index n. The automorphism group of a finite cyclic group of order n is isomorphic to with the isomorphism given by. This is often useful in understand fusion systems, where the pcore of automorphism groups is under good control, and so the gln, p part is the primary interest.
Lifting automorphisms of quotients by central subgroups ben kane. In 11, hillar and rhea give a useful description of the automorphism group of an arbitrary abelian p group, and they compute the size of this automorphism group. A special case of our theorem is that if p is an odd prime then autg cannot be isomorphic to the standard wreath product cp cp of two groups of order p, for any group g. The automorphism group of a generalized extraspecial p group. We begin with properties we have already encountered in the homework problems. Similarly, nontrivial powerful pgroups and groups of class 3 with cyclic centre are excluded from having this property corollary 5. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime, or twice a power of an odd prime. Automorphism group of finite cyclic group theorem aut.
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. Let t be the automorphism group of a nonelementary reduced abelian p group, p 5. Every cyclic group is an abelian group meaning that its group operation is commutative, and every finitely generated abelian group is a direct product of cyclic groups.
The automorphism group of an object x in a category c is denoted aut c x, or simply aut x if the category is clear from context. Pdf splitting the automorphism group of an abelian p. Access full article top access to full text full pdf how to cite top. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Automorphisms of p groups with cyclic commutator subgroup. It has been shown that there is a hamilton cycle in every connected cayley graph on any group gwhose commutator subgroup is cyclic of primepower order. For such a surface b, autb denotes the group of regular isomorphisms on b, or equivalently the group of biholomorphic maps on the complex surface b. Therefore, the action of a by conjugation inverts g. In general, for an elementary abelian group of order, the automorphism group is the general linear group. It is well known that if g is a finite noncyclic abelian p group of.
Rendiconti del seminario matematico della universita di padova 1993 volume. For a finite cyclic group of order, the automorphism group is of order where denotes the euler totient function. Pdf commuting automorphism of pgroups with cyclic maximal. The automorphism group of a split metacyclic pgroup. The size of an automorphism group wolfram demonstrations. Given a field extension, the automorphism group of it is the group consisting of field automorphisms of l that fixes k. The automorphism group of a finite minimal nonabelian pgroup. In particular g is simple if and only if the order of g. The main idea was to use the examples of lie algebras with derivation algebra of smaller dimension, from which they constructed a family of examples of pgroups with small automorphism group. Finite abelian and abelian automorphism group implies cyclic. Automorphism groups of abelian pgroups jutta hausen1 abstract. Suppose is a finite abelian group that is also a group whose automorphism group is abelian, i. Automorphism groups with cyclic commutator subgroup.
Since the elements of a cyclic group are the powers of an element, properties of cyclic groups are closely related to the properties of the powers of an element. Inner and outer automorphism, definition and related theorems commutator of a group, definition and related theorem derive group or commutative group, definition and related theorem. What is the automorphism group of the additive group of the p. The full automorphism group of a cyclic pgonal surface. If gis cyclic of order n, then i must be careful to map the generator ato an element that is killed by n. Further, the automorphism group is cyclic iff is 2,4, a power of an odd prime. The automorphism group of finite pgroups sciencedirect. Abstract we study the automorphism groups of finite, non abelian, 2generat ed pgroups with cyclic commutator subgroup, for odd primes p. Note that by autb we do not mean the birational automorphism group of b. In particular, we find a condition which guarantees all p,ngonal groups are conjugate in the full automorphism group of a p,ngonal surface, and we find an upper bound for the size of the.
The automorphism group of an abelian pgroup and its normal p. G that is bijective and satis es fxy fxfy for all x. I the set of automorphisms of g forms a group under function composition. Wreath products of cyclic pgroups as automorphism groups. The automorphism group of the cycle of length nis the dihedral group dn of order 2n. In this paper we study the automorphism group of a nite minimal nonabelian p group.
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