Maximal subgroup of p-group pdf

A maximal subgroup of a pgroup is always normal so that if a pgroup has an abelian subgroup of index p then this subgroup is a normal abelian subgroup. Then pjjnj, so n must be a pgroup, and therefore is a psylow subgroup of g. A group is a pgroup if the order is a prime or prime power. Pdf on the deskins index complex of a maximal subgroup. Some remarks on subgroups of maximal class in a finite pgroup. This means the sylow subgroup is a subgroup of g which is a pgroup, and is not a proper subgroup of any other psubgroup of g. A subgroup of order pk for some k 1 is called a psubgroup. Finite pgroups which have a maximal subgroup is full.

In conclusion we study, in some detail, the pgroups containing an abelian maximal subgroup. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. Maximal subgroups of finite groups university of virginia. Finally, we have proved two interesting theorems about nilpotent subgroup.

In what follows let r, s denote any nonempty subsets and h any subgroup of a. As mentioned before, in what follows we say that an element in a pgroup is uniform whenever its centralizer has order p 2. In 55 every elementary abelian 2subgroup of fi0 24 is shown to support an invariant symplectic form. A finite p group is a finite group whose order is a power of the prime,p. Thus we can translate the problem to gr which completes the proof. Pdf finite groups with nonnilpotent maximal subgroups. It is the intersection of all subgroups, where is maximal in. We obtain several results on the normal index of maximal subgroups m of composite index in g with g. Denote by dg the number of elements in a minimal generatorsystem of g and by.

Commutator subgroup and abelian quotient group problems. We do not yet know its size is the biggest ppower in jgj. Group actions, pgroups, and the sylow theorems mathematics 581, fall 2012 in this note we introduce the notion of a group action on a set and use it to prove a number of theorems about pgroups and the sylow theorems. G of the maximal pextension of f has a finite normal series with abelian factor groups. In what follows let r, s denote any nonempty subsets and h any subgroup of a group g. On the wielandt subgroup in a p group of maximal class. If g 14 then the only possible orders for a subgroup are 1, 2, 7 and 14. The set of all sylow psubgroups for a given prime p is sometimes written. A multiedge spinal group is a subgroup of the automorphism group of a regular padic rooted tree, generated by one rooted automorphism and a. Notation and terminology that will arise are briefly described below. We say that is a normal subgroup of group of prime power order if is a group of prime power order and is a normal subgroup of examples view.

We also study the noncyclic pgroups containing only one normal subgroup of a. For primes p and q, every group of order paqb is solvable. The frattini subgroup is the intersection of all maximal subgroups of g. Abelian p group corresponding to a p primary part of g is the direct product of cyclic groups. Maximal subgroups of multiedge spinal groups theofanis alexoudas, benjamin klopsch, and anitha thillaisundaram abstract. Two theorems about nilpotent subgroup lijiang zeng. Prove that gis cyclic of order pn for some prime p. If gis a pgroup and ma maximal subgroup of gthen mis normal in gand jg.

On pgroups of maximal class as groups of automorphisms. For a prime number p, a sylow psubgroup sometimes psylow subgroup of a group g is a maximal psubgroup of g, i. Department of mathematics, zunyi normal college, zunyi, china. Basically, the sylow subgroups play a role analogous to the role played by the maximal unipotent subgroups. On the other hand, it is well known that if a pgroup possesses an abelian subgroup of index p2 then it also has normal abelian subgroups of index p2. Then m is cnormal in gif and only if secmis trivial. Furthermore, we introduce the fundamental subgroup of a pgroup of maximal class by specifying its importance in the general theory of. Note that a maximal subgroup of g is not maximal among all subgroups of g, but only among all proper subgroups of g. A subgroup of index p k of a finite pgroup g is called a kmaximal subgroup of g.

P 6 0 mod p, so jpjis the largest ppower dividing jgj. In i960 hobby in 14 had proven that a nonabelian group with a cyclic center could not be a frattini subgroup of a pgroup. Then the factor group gbqdc is a pgroup without nontrivial subgroups. Finite groups with few noncyclic subgroups ii algebra. The sylow theorems the sylow theoremsdescribe the psubgroups of a. Journal of algebra 92, 4480 1985 maximal subgroups of finite groups m. Minimal nonabelian and maximal subgroups of a finite pgroup. A pgroup g which has exactly one nontrlvlal second maximal subgroup is cycllc if p 2 and has a cyclic normal subgroup. Prove that there is a subgroup k gsuch that hgi\k 0 and g. To every nite pgroup one can associate a lie ring lg, and if gg0is elementary.

Throughout this thesis a pgroup will mean a finite pgroup. The wielandt subgroup of a group g, denoted by wg, is the intersection of the normalizers of all subnormal subgroups of g. A finite pgroup is a finite group whose order is a power of the prime,p. This is a pgroup of order smaller than g,soby induction it is nilpotent, say.

The minimal normal subgroup of a solvable group is. Finite p groups with few nonmajor k maximal subgroups. A group of order pk for some k 1 is called a pgroup. On pgroups having the minimal number of conjugacy classes of maximal size a. Since we know all sylow subgroups of a given group are isomorphic to each other. Finite groups with nonnilpotent maximal subgroups 427 lemma 2. To every nite p group one can associate a lie ring lg, and if gg0is elementary. Moreover, g has a normal abelian subgroup with procyclic factor group. Metacyclic pgroups, pgroups of maximal class, minimal nonabelian pgroups, absolutely regular pgroups, abelian maximal subgroups, maximal abelian normal subgroups. Normalizer of sylow subgroups and the structure of a. Sylow subgroups which are maximal in the universal locally. Is a maximal abelian subgroup a of a free profinite group f necessarily isomorphic to z, the profinite completion of 2. In this paper, the authors classify the finite pgroups with.

For a group g and its subgroup n, we show that n is normal and gn is an abelian group if and only if the subgroup n contain the commutator subgroup of g. Commuting automorphism of pgroups with cyclic maximal subgroups. The frattini subgroup of, denoted, is defined in the following equivalent ways. Obrien abstract a longstanding question is the following. On pgroups having the minimal number of conjugacy classes. The number of maximal subgroups of a cyclic group is equal to the number of prime divisors of its order while the number of maximal subgroups of an elementary abelian p group of rank ris pr 1 p 1. Minimal nonabelian and maximal subgroups of a finite pgroup 99 i b1, theorem 5. In this paper we show that any pgroup with cyclic maximal subgroup is an aggroup. Maximal abelian subgroups of free profinite groups by dan haran and alexander lubotzky. G we denote the number of conjugate classes of the noncyclic subgroups of g. Throughout this thesis a p group will mean a finite p group. Finite pgroups which have a maximal subgroup is fullnormal p2 qiangwei song and fangfang xue department of math. The subgroup p 1 is a characteristic maximal subgroup of p.

In this paper, the authors show that for a pgroup of maximal class g, either w i g. The number of maximal subgroups of a cyclic group is equal to the number of prime divisors of its order while the number of maximal subgroups of an elementary abelian p. Maximal subgroups of a given group g are also related to its primitive actions. Any maximal subgroup m of g which does not contain s must be normal of index p in g, since s is central, and therefore we obtain r m. A sylow psubgroup of g is a maximal psubgroup of g. Sylow and playing a major role in the theory of finite groups. In this paper we proved some theorems on normal subgroups, onnormal subgroup, minimal nonmetacyclic and maximal class of a pgroup g. Let p be the largest prime dividing the order of a group g. Now, under the assumption that all maximal subgroups of a sylow sub group p are weakly s semipermutable subgroups of n g p, we shall establish. Computing the maximal subgroups of a permutation group i.

I tried to search online by i cant get a complete proof. On the deskins index complex of a maximal subgroup of a finite group article pdf available in proceedings of the american mathematical society 1142. For this reason, maximal subgroups are sometimes called maximal proper subgroups. Scott department of mathematics, california institute of technology, pasadena, california 91125 and department of mathematics, university of virginia, charlottesville, virginia 22903 communicated by walter feit received february 15, 1983 what ingredients are necessary to describe all maximal. Hence in any nite group there is a unique maximal normal nilpotent subgroup, and every nilpotent normal subgroup lies inside this. In the case where the group is the general linear group, the maximal unipotent subgroup can be taken as the group of upper triangular matrices with 1s on the diagonal, for instance. In the next theorem we list some of the basic properties of pgroups of maximal class, see for example 6, lecture 3. Gnis solvable, if pjjgnjthen by induction gnhas a maximal subgroup of ppower index, and the pullback of this subgroup to gwill be a maximal subgroup of gwith the same ppower index in g. If jgj p mwhere pdoes not divide m, then a subgroup of order p is called a. Minimal nonabelian and maximal subgroups of a finite p group 99 i b1, theorem 5. For a maximal subgroup m of a finite group g, the normal index of m is the order of a chief. Normal subgroup of group of prime power order groupprops. We prove this by induction on the power m of the order pm of the p group.