The breadth bx o f an element x of a finite pgroup g is defined by the equation pbcz 1 g. Let hbe a subgroup of the group gwith the property that whenever two elements of gare conjugate, then the conjugating element can be chosen within h. Introduction if g is a group and x, y g, then x,y x lylxy is the commutator of x and y. 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. On profinite groups with commutators covered by nilpotent subgroups.
The commutator subgroup of the general linear group over a field or a division ring k equals the special linear group provided that. Commutators and commutator subgroups in profinite groups cristina acciarri and pavel shumyatsky abstract. At the beginning of chapter 2, we mentioned the concept of a pgroup. The size of the commutator subgroup of finite groups. Computing commutator length in free groups springerlink. Show that gis not the settheoretic union of the conjugates of h. However, under some restrictions, the commutator subgroup of a finite pgroup consists entirely of commutators. Pdf finite p groups with cyclic commutator subgroup and cyclic. Sorry for adding one more answer, but heres a simple argument based on using that if the center is large enough then the commutator map has too small image to cover the derived subgroup, and even showing that, in suitable varying finite groups, the commutator length can be unbounded. Central quotient versus commutator subgroup of groups.
On the centralizer and the commutator subgroup of an. If i do the same computation with the other elements in q, ill always get the original subgroup back. Denote by kg the set of symmetric elements of the group algebra kg with respect to an oriented classical involution. This is the same set as the original subgroup, so the veri. Pdf the size of the commutator subgroup of finite groups.
In this paper we consider an arbitrary finite p group with a commutator subgroup g of order p, where p is a prime. It is easy to prove that gg is free abelian, where here g denotes the commutator subgroup of g. If one considers a subset y of g which freely generates, modulo g. For p odd, such groups are represented as a central product of 2generator. If g, is a finite group and g its commutator subgroup derived group then every product of the n elements of g is in the same coset of g. Some properties of the autocommutator subgroup of a 2 group m. But what about the situation when we look at pgroups for an odd prime p.
Pdf the commutator subgroup and schur multiplier of a. Find the order of d4 and list all normal subgroups in d4. In other words, a subgroup n of the group g is normal in g if and only if gng. On finite alperin p groups with homocyclic commutator. Thus there is an easy characterization of pgroups of class 1. The commutator subgroup and schur multiplier of a pair of. We prove this by induction on the power m of the order pm of the p group. Finite p groups with cyclic commutator subgroup and cyclic center springerlink. A purely algebraic algorithm is constructed for computing commutator length in a free group f 2 thm. Thus, the corollary also describes the finite simple groups in which a sylow 2 sub group has an abelian subgroup of index 2.
Chapter 7 nilpotent groups recall the commutator is given by x,yx. The autocommutator subgroup groups with given autocommutator subgroup. Pdf finite p groups with a cyclic commutator subgroup. Spring 2010 problem 2 let gbe a nite group with commutator subgroup g0. The center of pis nontrivial and a subgroup of size pin zp is a normal subgroup of g, necessarily minimal. Abstract we study the automorphism groups of finite, non abelian, 2generat ed pgroups with cyclic commutator subgroup, for odd primes p. However, most pgroups are of class 2, in the sense that as n. A pgroup has the converse property that every element of the appropriate coset of g is expressible as the product of. In fact, all these groups have modular subgroup lattices. Assume gis abelian and t is the torsion subgroup of gi. The commutator subgroup and schur multiplier of a pair of finite pgroups article pdf available in journal of the australian mathematical society 8101 august 2006 with 73 reads. Math 120a fall 2007 hw8 solutions 107 section 14 problem 26. Abelian subgroup of index 2, then the commutator subgroup of the sylow 2subgroup is cyclic. The structure of the commutator subgroup of sylow 2subgroups of an alternating group a 2 k is determined.
By a commutator lengthclg of an element g in a derived subgroup g. In this short paper we will try to answer this question by giving a bound on the plength of a psolvable group in terms of some powercommutator conditions in the sylow psubgroup. Pdf in the paper, finitepgroupsg with cyclic subgroup and cyclic center are described up to isomorphism under the additional condition. On commutators in pgroups of maximal class and some. Cx\, where cx is the centralizer of x in g, so that pbcz is the number of distinct conjugates of x.
If p 3, we prove that every quasicore3 3group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. The commutator subgroup math 430 spring 2011 let g be any group. In the paper, finitepgroupsg with cyclic subgroup and cyclic center are described up to isomorphism under the additional condition forp2. As this example indicates, it is generally infeasible to show a subgroup is normal by checking the. How to show that the commutator subgroup is a normal subgroup.
The commutator subgroup of the alternating group a 4 is the klein four group. Then gis solvable if and only if the factors in its composition series are cyclic of prime order. Hot network questions what could cause a spike in the natural uranium depletion rate. The abelian type invariants of the commutator quotient group gg. Finite p groups with cyclic commutator subgroup and. Abelian p group corresponding to a p primary part of g is the direct product of cyclic groups. Each gi is a normal subgroup of g and hence of each earlier subgroup in the chain by exercise ii. In this talk, we show that this is always the case if the commutator subgroup can be generated by 2 elements, thus generalising a result of guralnick who reached the same conclusion with the extra assumption that the group is metabelian. Notice, however, that there is a largest possible class for each order.
Then there exists a group g with g k 4 fg if and only if. The commutator and centralizer description of sylow 2subgroups of alternating and symmetric groups authors. This work continues the previous investigations of me, where minimal generating sets for sylow 2subgroups of alternating groups were constructed. We also show that the frattini subgroup of a quasicore2 2group is abelian. Automorphisms of pgroups with cyclic commutator subgroup. Commutator subgroup and abelian quotient group problems.
Also if h is any group other than an abelian twogenerator group, then for all sufficiently large abelian groups a, the wreath product a wr h provides an example where not every element in the commutator subgroup is a commutator. In i960 hobby in 14 had proven that a nonabelian group with a cyclic center could not be a frattini subgroup of a pgroup. Some properties of the autocommutator subgroup of a 2group. Let g be a psolvable group and p a sylow psubgroup of g. The pgroup p contains a subgroup of index p, and its direct product with his a subgroup of gwith index p, which means it must be maximal. The derived subgroups of g produce a subgroup chain g g1 g2.
Rogalski 1 some qual problems and some techniques spring 2008, 1 let gbe a nite group and ha proper subgroup. Solutions of some homework problems math 114 problem set 1 4. Moreover, if g is a prop group that has m procyclic subgroups whose union contains all commutators, then g is either fi nite of mbounded. The usual notation for this relation is normal subgroups are important because they and only they can be used to construct quotient groups. The set of all necklaces disregarding symmetry is a set xand we want to the number of orbits under the action of d.
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. It is known that, if the minimum number dg of generators of a finite alperin pgroup g is n. Now let g be an arbitrary finite pgroup or infinite topological prop group with derived subgroup g. Advances in mathematics 45, 319330 1982 commutators and commutator subgroups robert m. Commutators and commutator subgroups sciencedirect. Show that the property of being a commutator is invariant under conjuation in fact it is invariant under all automorphisms. Groups with the autocommutator subgroup a p group finite abelian groups as. The commutator subgroup is generated by commutators. Pdf finite pgroups with a cyclic commutator subgroup are described. Prove that the commutator subgroup g0of gis contained in h.