The particular interest in the existence of a pgroup with abelian automorphism group is motivated by a cryptographic protocol proposed in. Coleman automorphisms of finite groups and their minimal normal subgroups. On inner automorphisms and certain central automorphisms. We prove that if g is a finite nonabelian pgroup such that g z g is powerful then g has a noninner automorphism of order p leaving either. If g is a nonabelian pgroup, then g has a noninner pautomorphism. Then every noncentral normal subgroup of y contains a noncentral elementary abelian normal psubgroup of y of rank at least 2. In this paper we propose the group of unitriangular matrices over a. On coleman automorphisms of finite groups and their.
With over a century of literature on group theory including many books, it could take a while to locate such information. Then ghas a noninner automorphism of ppower order inducing the identity on gm and m if and only if im. Sep 11, 2001 finite p groups with few p automorphisms finite p groups with few p automorphisms 20010911 00. Examples of non cyclic group with a cyclic automorphism group. Automorphisms which centralize a sylow psubgroup core. The group of autocentral automorphisms mohammad reza r. Automorphism group computation and isomorphism testing in. Some remarks on commuting fixed point free automorphisms of. A5 is a centerless group, it embeds as a subgroup of index two inside its automorphism group, which is symmetric group on five elements. There has been a number of results on the central automorphisms of a group. A simple generalization of elgamal cryptosystem to nonabelian groups ayan mahalanobis department of mathematical sciences, stevens institute of technology, hoboken, nj 07030. From now on g will stand for a purely nonabelian pgroup of class 2, where p is an. For g a noncyclic pgroup of order pn with n 3, jgjdivides jautgj.
A short proof of existence of noninner automorphism of. Noninner automorphisms of order p in finite normally. A p group gis said to be extraspecial if g0 zg g is of order p. The topic of pgroups with abelian automorphism groups abelian autg has inter ested researchers for years. Assume n is a finite pgroup on which a finite abelian p. In this paper it is shown that if g is a finite nonabelian metacyclic. Some finite pgroups with central automorphism group of. A conjecture of berkovich asserts that every non simple finite p group has a non inner automorphism of order p. Prove the set of all automorphisms of a group is a group.
Powerful pgroups have noninner automorphisms of order p and. In on groups with a classpreserving outer automorphism by brooksbank and mizuhara, the following is said. We prove that a group of order p3 having a commuting fpf automorphism, has a restricted structure. If gis a p group then the centre of gis a non trivial. Coleman automorphisms of finite groups and their minimal. For some groups every automorphism is an inner automorphism. Finite pgroups without noninner automorphism of order p and the existence of noninner automorphism of order p in powerful pgroups let g be a group and let a be a normal abelian subgroup ofg. Let g be a finite simple nonabelian group and let p be an.
This research was in part supported by a grant from ipm no. Also, a nonabelian group 2010 mathematics subject classi cation. Also, a non abelian group 2010 mathematics subject classi cation. The center of the multiplicative group of nonzero quaternions is the multiplicative group of nonzero real numbers. Finite p groups with few p automorphisms finite p groups with few p automorphisms 20010911 00. We also recall a connection between the conjecture and. The fact that acgjag is not trivial can either be viewed as a special instance where g admits non inner 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. Using the class equation one can prove that the center of any nontrivial finite pgroup is nontrivial. Inner automorphisms are about the only examples of automorphisms that can be written down without knowing extra information about the group such as being told the group is abelian or that it is a particular matrix group. On inner automorphisms and certain central automorphisms of groups springerlink. Some nonabelian 2groups with abelian automorphism groups. Some finite p groups with central automorphism group of non minimal order journal of algebra and its applications.
We describe the automorphism group auta using the rank of a and its torsion part ppart a p. Every automorphism of g induces an automorphism on g. As remarked earlier, it is enough to show that g is solvable. But in a pgroup any proper subgroup has strictly bigger normalizer. In this paper we study the longstanding conjecture of whether there exists a non inner automorphism of order p for a finite non abelian pgroup. The particular interest in the existence of a p group with abelian automorphism group is motivated by a cryptographic protocol proposed in 10. 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. If g is a pgroup then the centre of g is a nontrivial subgroup of g. Liebeck 8 has shown that finite p groups of class 2 with p 2 must have a noninner automorphism of order p fixing the frattini subgroup elementwise. Automorphisms of finite groups inder bir singh passi.
These groups provide counterexamples to a conjecture of a. Solutions of some homework problems math 114 problem set 1 4. On central automorphisms of finite p groups request pdf. Orfi 12 for the finite nonabelian pgroups with cyclic. Finite groups with fixedpointfree automorphisms of prime. If we take to be the inner automorphisms of g by the elements of p we obtain t. One can also read o from the pims that whas zero 1cohomology whenever p 7. Automorphism group computation and isomorphism testing in nite groups john j. Also we prove that autg has a noninner automorphism of order 2 when g dn or qn and n is power of 2.
We construct a family of nonspecial finite pgroups having abelian automorphism 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. We remark that the proof of the above also tells us that if im. A result of wolfgang gaschutz says that if g is a finite nonabelian pgroup, then g has an automorphism of ppower order which is not inner. Some finite pgroups with central automorphism group of nonminimal order journal of algebra and its applications. Moghaddam and hesam safa faculty of mathematical sciences and centre of excellence in analysis on algebraic structures, ferdowsi university of mashhad, iran group theory conference, mashhad 2010 ferdowsi university of mashhad, march 1012, 2010, p. On coleman automorphisms of finite groups and their minimal normal subgroups arne van antwerpen june 19, 2017. Researchers are working on protocols using nonabelian groups for secret sharing or exchange of private keys over a nonsecure channel. Liebeck 8 has shown that finite pgroups of class 2 with p 2 must have a noninner automorphism of order p fixing the frattini subgroup elementwise. A pgroup gis said to be extraspecial if g0 zg g is of order p. This conjecture is far from being proved despite the great effort devoted to it. If g is a purely nonabelian pgroup, then autcentg is also a pgroup. The automorphism group of a finite metacyclic pgroup richard m. Notice that each non central element gof ginduces a non trivial automorphism of gvia conjugation.
Author links open overlay panel arne van antwerpen 1. It is an open problem whether every non abelian p group g has an automorphism of order p. Automorphism group computation and isomorphism testing. Automorphism groups of finite p groups of coclass 2 deepdyve.
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. Autg containing inng, the group of inner automorphisms of g. Find the order of d4 and list all normal subgroups in d4. Throughout this paper all groups are assumed to be. A simple generalization of elgamal cryptosystem to non. Noninnerautomorphisms powerful pgroups pcentralgroups in this paper we study the longstanding conjecture of whether thereexistsanoninner automorphism oforder p fora. Since ig is an invertible homomorphism, its an automorphism. A short proof of existence of noninner automorphism of order.
We also construct a family of finite pgroups having nonabelian automorphism. If we leave out the condition of being subgroupclasspreserving, then the above script indeed finds a noninner automorphism that is classpreserving, though i havent checked if its the same one. A nonabelian group of primepower order is said to have divisibility property if its order divides that of. Showing that a cyclic automorphism group makes a finite. Jun 15, 2008 let a be a finitely generated abelian group. Homework equations to prove that this is a group i must show that it is closed on composition, there. 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, for any prime equation. For finite groups, being abelian and the automorphism group being abelian as well implies cyclic. In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order p for a finite nonabelian pgroup. If gis a pgroup then the centre of gis a nontrivial subgroup of g. The automorphism group of a finite metacyclic pgroup. Holt2 1school of mathematics, university of sydney, sydney, australia 2mathematics institute, university of warwick, coventry, great britain abstract a new method for computing the automorphism group of a nite permu. The set of inner automorphisms of gis denoted inng.
If the quotient group gzg is cyclic, g is abelian and hence g zg, so gzg is trivial. Conjecture every finite pgroup admits a noninner automorphism of order p, where p denotes a prime number. In this paper we study the existence of at least one noninner automorphism of order p in a finite normally constrained pgroup when p is an odd prime. When is the automorphism group of a finite p group abelian. 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.
It is shown that if g is a finite pgroup of coclass 2, then the order of g divides the order of the automorphism group of g. We characterize finite p groups of order up to p7 whose central automorphisms fixing the center elementwise are all inner. Curran and mccaughan 4 proved that for any nonabelian. Let g be a finite pgroup and denote by x,g the members of the lower. We continue with some basic facts about the automorphism group of m. It is an open problem whether every nonabelian pgroup g has an automorphism of order p. Noninner automorphisms of order p in finite pgroups of coclass 3. In particular g is simple if and only if the order of g. Homework statement an isomorphism of a group onto itself is called an automorphism. Is an automorphism of a finite group inner when it. In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order p for a finite non abelian pgroup. A simple generalization of elgamal cryptosystem to nonabelian groups.
The group g0of all inner automorphisms is isomorphic to gz. We prove that if g is a nite nonabelian pgroup such that g zg is powerful then g has a noninner automorphism of order p leaving either g or 1zg elementwise xed. I s, then g has no noninner coleman automorphism, and thus, the normalizer problem holds for g. Jafari1 department of mathematics shahrood university of technology p. We characterize finite pgroups of order up to p7 whose central automorphisms fixing the center elementwise are all inner. Finite pgroups with the least number of outer pautomorphisms. Let g be a finite group with a fixedpointfree automorphism of prime ler. Prove that the set of all automorphisms of a group is itself a group with respect to composition.
Some finite pgroups with central automorphism group of non. A non abelian group of primepower order is said to have divisibility property if its order divides that of its automorphism group. The noninner automorphisms are called outer automorphisms. On finite p groups whose automorphisms are all central. Request pdf on central automorphisms of finite p groups we characterize all finite pgroups g of order pnn\leq 6, where p is a prime for n\leq 5 and an odd prime for n 6, such that the. Researchers are working on protocols using non abelian groups for secret sharing or exchange of private keys over a non secure channel. It is conjectured that in this case jgj jautgj in other words jgj c jautgj p. A result of wolfgang gaschutz says that if g is a finite non abelian p group, then g has an automorphism of ppower order which is not inner. Just as every finite pgroup has a nontrivial center so that the inner automorphism group is a proper quotient of the group, every finite pgroup has a nontrivial outer automorphism group. That is, in an abelian group the inner automorphisms are trivial. It is well known that if g is a finite noncyclic abelian pgroup of. The non inner automorphisms are called outer automorphisms.
Powerful pgroups have noninner automorphisms of order p. Non inner automor phisms of order p in finite p grou ps of coclass 3 springerlink. For a given prime a we find the smallest order of a non abelian pgroup admitting a commuting fixed point free automorphism. In particular gis simple if and only if the order of g. If g is a pgroup then the centre of g is a nontrivial. Since any automorphism permutes the conjugacy classes.
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