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. Automorphisms of pgroups with cyclic commutator subgroup. Advances in mathematics 45, 319330 1982 commutators and commutator subgroups robert m. 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. By a commutator lengthclg of an element g in a derived subgroup g. Let g be a psolvable group and p a sylow psubgroup of g. Then gis solvable if and only if the factors in its composition series are cyclic of prime order. 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. 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. The commutator subgroup math 430 spring 2011 let g be any group. Hot network questions what could cause a spike in the natural uranium depletion rate. Next groups with small abelian commutator subgroups are studied. 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. Abstract we study the automorphism groups of finite, non abelian, 2generat ed pgroups with cyclic commutator subgroup, for odd primes p.
A pgroup has the converse property that every element of the appropriate coset of g is expressible as the product of. The abelian type invariants of the commutator quotient group gg. In annals of discrete mathematics, 1991 4 super pgroups. The center of pis nontrivial and a subgroup of size pin zp is a normal subgroup of g, necessarily minimal. 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 set of all necklaces disregarding symmetry is a set xand we want to the number of orbits under the action of d. We prove this by induction on the power m of the order pm of the p group. Automorphism groups with cyclic commutator subgroup. Commutators and commutator subgroups in profinite groups cristina acciarri and pavel shumyatsky abstract. Finite p groups with cyclic commutator subgroup and cyclic center springerlink. 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. Pdf in the paper, finitepgroupsg with cyclic subgroup and cyclic center are described up to isomorphism under the additional condition. Some properties of the autocommutator subgroup of a 2 group m.
A purely algebraic algorithm is constructed for computing commutator length in a free group f 2 thm. On the centralizer and the commutator subgroup of an. The commutator subgroup is generated by commutators. We also show that the frattini subgroup of a quasicore2 2group is abelian. The commutator subgroup of the alternating group a 4 is the klein four group. Solutions of some homework problems math 114 problem set 1 4. It is known that, if the minimum number dg of generators of a finite alperin pgroup g is n. Pdf finite p groups with a cyclic commutator subgroup.
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. In this paper we consider an arbitrary finite p group with a commutator subgroup g of order p, where p is a prime. Pdf finite pgroups with a cyclic commutator subgroup are described. Thus there is an easy characterization of pgroups of class 1. Abelian subgroup of index 2, then the commutator subgroup of the sylow 2subgroup is cyclic.
However, most pgroups are of class 2, in the sense that as n. Pdf the commutator subgroup and schur multiplier of a. Minimum number of swaps in sorting sequence numbers poetics 1 and 2019. The structure of the commutator subgroup of sylow 2subgroups of an alternating group a 2 k is determined. Assume gis abelian and t is the torsion subgroup of gi. The usual notation for this relation is normal subgroups are important because they and only they can be used to construct quotient groups.
But what about the situation when we look at pgroups for an odd prime p. The commutator subgroup of the general linear group over a field or a division ring k equals the special linear group provided that. Next, we prove that if g is a wreath product g a p, with a a nontrivial finite abelian group and p a pgroup of maximal. Pdf the size of the commutator subgroup of finite groups. Thus, the corollary also describes the finite simple groups in which a sylow 2 sub group has an abelian subgroup of index 2.
However, under some restrictions, the commutator subgroup of a finite pgroup consists entirely of commutators. In i960 hobby in 14 had proven that a nonabelian group with a cyclic center could not be a frattini subgroup of a pgroup. The size of the commutator subgroup of finite groups. Notice, however, that there is a largest possible class for each order. Abelian p group corresponding to a p primary part of g is the direct product of cyclic groups.
Commutators and commutator subgroups sciencedirect. As this example indicates, it is generally infeasible to show a subgroup is normal by checking the. This is the same set as the original subgroup, so the veri. 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. Commutator subgroup and abelian quotient group problems. Math 120a fall 2007 hw8 solutions 107 section 14 problem 26. 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. At the beginning of chapter 2, we mentioned the concept of a pgroup. If one considers a subset y of g which freely generates, modulo g. Prove that the commutator subgroup g0of gis contained in h. In the paper, finitepgroupsg with cyclic subgroup and cyclic center are described up to isomorphism under the additional condition forp2. 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. Show that the property of being a commutator is invariant under conjuation in fact it is invariant under all automorphisms. The derived subgroups of g produce a subgroup chain g g1 g2.
On commutators in pgroups of maximal class and some. Mathematics free fulltext the derived subgroups of. Central quotient versus commutator subgroup of groups. Cx\, where cx is the centralizer of x in g, so that pbcz is the number of distinct conjugates of x. How to show that the commutator subgroup is a normal subgroup. Chapter 7 nilpotent groups recall the commutator is given by x,yx. Computing commutator length in free groups springerlink. Pdf finite p groups with cyclic commutator subgroup and cyclic. Find the order of d4 and list all normal subgroups in d4. Now let g be an arbitrary finite pgroup or infinite topological prop group with derived subgroup g. On finite alperin p groups with homocyclic commutator.
In fact, all these groups have modular subgroup lattices. The autocommutator subgroup groups with given autocommutator subgroup. Show that gis not the settheoretic union of the conjugates of h. Finite p groups with cyclic commutator subgroup and. The commutator and centralizer description of sylow 2subgroups of alternating and symmetric groups authors. On profinite groups with commutators covered by nilpotent subgroups.
The breadth bx o f an element x of a finite pgroup g is defined by the equation pbcz 1 g. It is easy to prove that gg is free abelian, where here g denotes the commutator subgroup of g. Groups with the autocommutator subgroup a p group finite abelian groups as. 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. Rogalski 1 some qual problems and some techniques spring 2008, 1 let gbe a nite group and ha proper subgroup. Spring 2010 problem 2 let gbe a nite group with commutator subgroup g0.
For p odd, such groups are represented as a central product of 2generator. If i do the same computation with the other elements in q, ill always get the original subgroup back. In other words, a subgroup n of the group g is normal in g if and only if gng. Then there exists a group g with g k 4 fg if and only if. This work continues the previous investigations of me, where minimal generating sets for sylow 2subgroups of alternating groups were constructed. Some properties of the autocommutator subgroup of a 2group. Converse of the schurs theorem is not true in general as shown by infinite extraspecial p groups, where p is an odd prime. The commutator subgroup and schur multiplier of a pair of. Introduction if g is a group and x, y g, then x,y x lylxy is the commutator of x and y.