Then there exists a pgroup g with nilpotency class 2 such that, the set of conjugacy class sizes of g is exactly. In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal psubgroup ng of the automorphism group, autg of a bounded or finite abelian pgroup g. Corollary 2 let g be a nite 2group with conjugate type 1. Part i concerns the following counting function and various refinements. The qtensor square of a powerful pgroup pdf free download.
In any nilpotent group the lower and upper central series see subgroup series breaks off at the trivial subgroup the group itself, and their lengths are equal to the nilpotency class of the group. For a prime p and a finite pgroup p, we define the coexponent of p to be the least integer fp such that p possesses a cyclic subgroup of index p fp, and the purpose of this paper is to derive a bound for the nilpotency class in terms of the coexponent. The coexponent of a finite p group is introduced and we consider how the nilpotency class is bounded in terms of this invariant. We consider the problem of how the nilpotency class. The coexponent of a finite pgroup is introduced and we consider how the nilpotency class is bounded in terms of this invariant. The nilpotency class of the unit group u of a modular pgroup algebra fg is determined when p is odd and ghas a cyclic commutator subgroup. If every p element of ngp commutates with every element of p and the nilpotent class of p is less p. For more information of the number of pgroups of various nilpotency class values for various primes, refer nilpotency class distribution of pgroups. On 2symmetric words in nilpotent groups springerlink.
The least integer satisfying this condition is called the nilpotency class of. Nilpotency degree of integral cohomology classes of p. Michaux state forest bureau of forestry michaux state forest 34 174 696 997 997 233 234 394 233 carlisle. The nilpotency class of pgroups in which subgroups have few. This is the first part of two papers devoted to compute the nilpotency class of. Transporting cohomology in lazard correspondence 3 algebra and the zplie subalgebra lxp generated by xis the free nilpotent zp lie algebra of nilpotency class p. In the case q 0, moravec showed that if g is a group of nilpotency class. Also, a nonabelian group 2010 mathematics subject classi cation. Then the nilpotence class of l is bounded in terms of. On pgroups having the minimal number of conjugacy classes. The 2groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups. Nilpotency class of symmetric units of group algebras. Muller 16, theorem has shown that if g is neither elementary abelian nor extraspecial, then aut. The purpose of this study is to compute the epicenter of finite nonabelian metacyclic pgroups of nilpotency class two, for some small order groups, using groups, algorithms and programming gap.
The nilpotency class of pgroups in which subgroups have. We consider the problem of how the nilpotency class of a finite pgroup can be bounded in terms of the maximum length of the conjugacy classes of cyclic subgroups. Since zg is a nitely generated abelian group, our task. Then the nilpotency class of g is bounded in terms of. Since autcg inng for all abelian groups g, it is su. If g is a finite group and p is a sylow psubgroup of g then. For the proof of the following well known result, one can see 26, pg 69. The semidihedral group also denoted is the semidihedral group also called quasidihedral group of order. Nilpotency class of a p group of order pa is at most a1.
Any abelian group is a group of nilpotency class two in fact, it has class one. Computing the nonabelian tensor square of metacyclic pgroups of nilpotency class two. Since groups of order p p are always regular groups, it is also a minimal such example. W e note that a pgroup of order p n with a conjugacy class of length p n. We will discuss results that bound the nilpotency class of a powerful pgroup in terms of the exponent of a quotient by a normal abelian subgroup. This is the first part of two papers devoted to compute the nilpotency class of ng using formulas, and algorithms that work in almost all groups.
Now, we can modify problem 3, and state it as problem 3 find a bound on the nilpotency class of pgroups with exactly two conjugacy class sizes, for odd primes p. We find the nilpotency class of a group of 2symmetric words for free nilpotent groups, free nilpotent metabelian groups, and free nilpotent of class cbyabelian groups. Finite pgroups with the least number of outer pautomorphisms. In particular, we prove that every finite pgroup is nilpotent for a prime p. If g is a nilpotent group then the nilpotency class of g is the smallest n. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Let p be a prime number and let g be a pgroup which is not elementary abelian. The qtensor square of a powerful pgroup sciencedirect. This group is the external direct product of the quaternion group of order eight and the cyclic group of order two arithmetic functions want to compare and contrast arithmetic function values with other groups of the same order. By the equivalence of definitions of finite nilpotent group, every finite nilpotent group is a direct product of its sylow subgroups.
The nilpotency class of finite groups of exponent p. For instance, if g is abelian, then it has class 1. A survey 4 in this part we consider pgroups of small orders and mention results on class preservingautomorphismsof pgroupsoforderat most p5. A nilpotent group can be thought of a group that is only finitely removed from an abelian group. On pgroups of conjugate rank 1 and nilpotency class 3. A bound for the nilpotency class of a finite pgroup in. The length of a shortest central series of a nilpotent group is called its class or degree of nilpotency. A pgroup of order p and nilpotency class c is said to have coclass rn c. We also provide the size of the conjugacy class of noncentral elements of a group g in v fg. My guess is no, at least as regards a classification. In this video, we prove that any pgroup is nilpotent and the nilpotency class of a pgroup of order p power a is at most a1 for a more than 2. A pgroup g of maximal nilpotency class and order at least p4 has p. In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal psubgroup.
In the case q 0, moravec 12 showed that if g is a group of nilpotency class. The multiplicative group of upper unitriangular n x n matrices over any field f is a nilpotent group of nilpotency class n 1. Examples of groups of larger nilpotency class and satisfying hypothesis a are given below. A group of order, with prime, can have any nilpotency class between 1 and if. This is a p group of order smaller than g,soby induction it is nilpotent, say. Furthermore, every finite nilpotent group is the direct product of pgroups. This is a pgroup of order smaller than g,soby induction it is nilpotent, say. A group g of nilpotency class c may have a central series of length greater than c. Pdf on the nilpotency class of a group of exponent four. We find the nilpotency class of a group of 2symmetric words for free. Recall that if gis a group then zg fa2g jab bafor all b2gg.
Pdf conjugacy classes and characters of finite p groups. This is done via an extension of a theorem of coleman and. In particular, all 3groups with maximal nilpotency class and order at least 34 have 6 conjugacy classes of maximal size 2, 14. Using transfinite recursion, the notion of nilpotency class can be extended to any ordinal.
G is a normal sylow psubgroup of autg, when g is an extraspecial pgroup. Independent of the nilpotency class, this result is also true if we consider a powerful finite pgroup cf. This links into the previous proposition via the following lemma. Wright and others published on the nilpotency class of a group of exponent four find, read and cite all the research you need on researchgate. Various other topics can be investigated by using presentations. We sharpen some previously known bounds and also prove that a pgroup in which every cyclic subgroup has at most p 2 conjugates has class. Pdf in this paper, we present a sharp bound for the nilpotency class of a finite pgroup where p is an odd prime in terms of its coexponent. A new algorithm for finding the nilpotency class of. In 1, bacon and kappe give a classification of 2generator pgroups of nilpotency class 2 with the goal of computing their. A bit wild natural problem which arises here 21, problem 6. Burnside theorem and finite pnilpotent groups iazd. Pdf computing the nonabelian tensor square of metacyclic. Specifically, it is a group such that is the trivial group, for some integer, where is the th term of the lower central series of.
The trivial group is a group of nilpotency class two in fact, it has class zero. The coclass graph associated with pgroups of maximal class heiko dietrich tu braunschweig, germany maximal class and coclass a pgroup of order pn has maximal class if its nilpotency class is n. In this paper we calculate the class of a free group of finite rank of the product of two varieties, where the left factor is the variety of all abelian groups of exponent pn and the right is the variety of all nilpotent groups of class at most c and exponent pt where c nilpotency class c is bounded by a function depending only on p, c and d. If gis a nilpotent group then the nilpotency class of gis the smallest n 0 such that z ng g. Now we restrict our attention to establishing some bounds for exponents. The group can also be defined as the general semilinear group of degree one over the field of nine elements arithmetic functions.