Minimal nonabelian and maximal subgroups of a finite pgroup 99 i b1, theorem 5. Our next result shows that psoluble groups with sylow psubgroups isomorphic to a normal subgroup of a minimal nonypgroup have a. Janko, in preparation about finite pgroups with many minimal nonabelian subgroups is solved. Metacyclic p groups, p groups of maximal class, minimal nonabelian p groups, absolutely regular p groups, abelian maximal subgroups, maximal abelian normal subgroups. Assume by contradiction that gis a counterexample of minimal order, h6gis chosen so that. This focuses attention on the structure of pgroups and the automorphisms of pgroups.
We prove this by induction on the power m of the order pm of the p group. First we show that a is extendible to the preimage a in g of any minimal normal subgroup a of g. Pdf on p nilpotency and minimal subgroups of finite groups. Coleman automorphisms of finite groups and their minimal. The cyclic subgroup of order two is not a normal subgroup. In conclusion we study, in some detail, the p groups containing an abelian maximal subgroup. As g is solvable, a is an elementary abelian group. The group has a minimal normal subgroup, and by 1 this subgroup is a p group for some prime p. Probabilistic generation of nite groups with a unique minimal normal subgroup, joint with a. Department of mathematics university of science and technology of china hefei 230026, p. This means that if h c g, given a 2 g and h 2 h, 9 h0,h00 2 h 3 0ah ha and ah00 ha. As n is the unique minimal normal subgroup, it follows that o p.
If, is the unique nontrivial proper normal subgroup, and contains only one other nontrivial normal subgroup the klein fourgroup. There is a classification of p groups with cyclic center and cyclic commutator subgroup. Recall that a minimal subgroup of a nite group is a subgroup. Detomi, university of padova probabilistic generation of nite groups with a unique minimal normal subgroup st. If g is a pgroup, then so is gz, and so it too has a nontrivial. When a chief factor minimal normal subgroup of a quotient group of a finite solvable group is contained in the frattini, then it is called a frattini factor, otherwise it is called complemented, since the quotient group is a semidirect product with normal kernel that chief factor, and complement a non normal maximal subgroup. Note that for 3, if g is a p group then it splits over each normal subgroup. Volume 224, issue 2, 15 february 2000, pages 198240. This kind of researches on groups with the minimal condition on subgroups not. Minimal nonabelian and maximal subgroups of a finite pgroup. Minimal normal subgroups of a finite group mathoverflow. While doing so, we regard an elementary abelian pgroup h as a vector space over z the field of p. In another direction, every normal subgroup of a finite p group intersects the center nontrivially as may be proved by considering the elements of n which are fixed when g acts on n by conjugation.
Influence of normality conditions on almost minimal subgroups. On second minimal subgroups of sylow subgroups of finite. For further discussion of the interesting problem of determining the structure of such minimal nonnilpotent groups, perhaps it su. Denote by o pg the maximal normal p subgroup of a group g. G pe, where e is a 2group with a normal subgroup f such that f. The group has a minimal normal subgroup, and by 1 this subgroup is a pgroup for some prime p. In this paper, we investigate the class of groups of which every maximal subgroup of its sylow p subgroup is c normal and the class of groups of which some minimal subgroups of its sylow p subgroup is c normal for some prime number p.
In this paper we proved some theorems on normal subgroups, onnormal subgroup, minimal nonmetacyclic and maximal class of a pgroup g. Journal of algebra 358 2012 178188 179 said to be minimal nonabelian if every proper subgroup of g is abelian. If n 6 g n normal subgroup of g, then we write n e g n. In this paper we proved some theorems on normal subgroups, on normal subgroup, minimal nonmetacyclic and maximal class of a pgroup g. Finite pgroups with a minimal nonabelian subgroup of. Apr 09, 2012 all finite nonabelian solvable groups have at least one normal group the commutator and therefore contain a minimal normal subgroup. Chapter 7 nilpotent groups recall the commutator is given by x,yx.
Let 1 np be a sylow p subgroup of n, and let q be a subgroup of g of order p2 such that q. Some known facts about minimal nonabelian pgroups are. Since every central subgroup is normal, it follows that every minimal normal subgroup of a finite p group is central and has order p. We also study the noncyclic pgroups containing only one normal subgroup of a. Abelian p group corresponding to a p primary part of g is the direct product of cyclic groups.
It can be shown that a nite group is nilpotent if and only if it possesses a central series. Metacyclic pgroups, pgroups of maximal class, minimal nonabelian pgroups, absolutely regular pgroups, abelian maximal subgroups, maximal abelian normal subgroups. If some minimal normal subgroup of g were to avoid the frattini subgroup, it would be a direct factor, so the socle must lie in the frattini subgroup. School of mathematics and statistics mt5824 topics in. In a nilpotent group, any minimal normal subgroup must actually be a minimal subgroup i. Finite group with cnormal or squasinormally embedded. Maximal subgroups of solvable groups have prime power. Also the pgroup for which the above estimate is attained, are described in great detail. In other words, it is a series such that each is normal in and is a minimal normal subgroup of. So the p groups with a unique normal minimal subgroup are exactly the p groups with cyclic center.
Then there exists a subgroup a\ normal in g with z group is a p group if and only if the group splits over each normal subgroup. Chief series is a normal series where each successive quotient is a minimal normal subgroup in the quotient of the whole group by the lower end. On minimal faithful permutation representations of finite. In other words, a subgroup n of the group g is normal in g if and only if gng. Let pbe a prime dividing jgjand n be a minimal normal subgroup, so in particular nhas primepower size the prime is not necessarily. A subgroup kof a group gis normal if xkx 1 kfor all x2g. This is related to, but different from, the frattini subgroup, which is the intersection of all maximal subgroups. While doing so, we regard an elementary abelian p group h as a vector space over z the field of p. Cmrd 2010 school of mathematics and statistics mt5824 topics in groups problem sheet vi. A plocal nite group consists of a nite pgroup s, together with a pair of categories which encode \conjugacy relations among subgroups of s, and which are modelled on the fusion in a sylow p subgroup of a nite group. On the probability of generating a monolithic group, joint with a. Subgroup families controlling plocal finite groups c. Assume there exists minimal normal subgroup itexnitex such. Let n be a minimal normal subgroup of g, and suppose that n is not a pgroup.
Hence, n is an elementary abelian pgroup for some prime p. Recall from last time that if g is a group, h a subgroup of g and g 2g some xed element the set gh fgh. Let a be a finite abelian pgroup, b an elementary abelian subgroup of a. Normal subgroups and homomorphisms stanford university.
The number of minimal nonabelian subgroups in a nonabelian finite pgroup containing a maximal normal abelian subgroup of given index is estimated. A subgroup h of a group g is called normal if gh hg for all g 2g. Abstract a finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. The jacobson radical is defined as the intersection of all maximal normal subgroups. All finite nonabelian solvable groups have at least one normal group the commutator and therefore contain a minimal normal subgroup. In particular, the trivial subgroups are normal and all subgroups of an abelian group are normal. Department of mathematics, xuzhou normal university xuzhou 221116, p. Wenbin guo department of mathematics, university of science and technology of china. A oneheaded group is a group with a unique maximal normal subgroup. The usual notation for this relation is normal subgroups are important because they and only they can be used to construct quotient groups. If a is a p group of automorphisms of the abelian pgroup p which acts trivially. For example if g s 3, then the subgroup h12igenerated by the 2cycle 12 is not normal. Pdf finite permutation groups with a transitive minimal.
Minimal normal subgroups of a finite supersolvable groups. E and ef is isomorphic to a quaternion group of order. Minimal nonnilpotent groups which are supersolvable arxiv. Assume there exists minimal normal subgroup itexnitex such that itexn ot\subseteq hitex. If there exists a sylow p subgroup p of g such that every maximal subgroup of p is weakly ssemipermutable in ngp and p is. In conclusion we study, in some detail, the pgroups containing an abelian maximal subgroup. Probabilistic generation of finite groups with a unique. Normalizer of sylow subgroups and the structure of a.
Normal subgroups and factor groups normal subgroups if h g, we have seen situations where ah 6 ha 8 a 2 g. In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. Cernikov studied groups with the minimal condition on nonabelian sub groups, on non normal subgroups, on abelian non normal subgroups respectively. Indeed, the socle of a finite pgroup is the subgroup of the center consisting of the central elements of order p. Maximal subgroups of solvable groups have prime power index. Maximal subgroups of finite groups university of virginia. Let gbe a nite group and g the intersection of all maximal subgroups of g. Some remarks on subgroups of maximal class in a finite pgroup.
If m is a maximal subgroup of a group g and 77 is a minimal supplement to. We also study the noncyclic pgroups containing only one normal subgroup of a given order. Theminimalnormal psubgroupsof g areofthesame order, and it is at most p2. Finite pgroups with many minimal nonabelian subgroups.
We use conventional notions and notations, as in huppert 3. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as. A subgroup h of a group g is a normal subgroup of g if ah ha 8 a 2 g. I tried to search online by i cant get a complete proof. The set of transpositions is a minimal set of generators for. Since n is a minimal normal subgroup of a psolvable group g with opg 1, we deduce that p. The result is clear if jgjis a prime power in particular, if jgjis prime. Then nhas a complement in gif and only if n5 g solution assume that n has a complement h in g. The definition of a solvable group is given in gt3, section 1. Then there exists a subgroup a\ normal in g with z normal subgroups in d4. Introduction all groups considered in this paper will be nite. On the number of minimal nonabelian subgroups in a.
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