We also provide the size of the conjugacy class of noncentral elements of a group g in v fg. If g is a nilpotent group then the nilpotency class of g is the smallest n. Then the nilpotency class of g is bounded in terms of. On 2symmetric words in nilpotent groups springerlink. In the case q 0, moravec 12 showed that if g is a group of nilpotency class. Now we restrict our attention to establishing some bounds for exponents. For the proof of the following well known result, one can see 26, pg 69. This is done via an extension of a theorem of coleman and. A pgroup gis said to be extraspecial if g0 zg g is of order p. G of the automorphism group, autg of a bounded or finite abelian pgroup g. The least integer satisfying this condition is called the nilpotency class of.
Nilpotency class of a p group of order pa is at most a1. The qtensor square of a powerful pgroup pdf free download. A group of order, with prime, can have any nilpotency class between 1 and if. 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. Let p be a prime number and let g be a pgroup which is not elementary abelian. Pdf on the nilpotency class of a group of exponent four. 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.
Then the nilpotence class of l is bounded in terms of. If every p element of ngp commutates with every element of p and the nilpotent class of p is less p. Since the nilpotency class of a finite group of exponent p is the same as the nilpotency class of its associated lie ring, his theorem proves that the class of a finite mgenerator group of exponent p is bounded. On pgroups having the minimal number of conjugacy classes. Any abelian group is a group of nilpotency class two in fact, it has class one. Various other topics can be investigated by using presentations.
On pgroups of conjugate rank 1 and nilpotency class 3. The length of a shortest central series of a nilpotent group is called its class or degree of nilpotency. 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 the case q 0, moravec showed that if g is a group of nilpotency class. 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. A bit wild natural problem which arises here 21, problem 6. Furthermore, every finite nilpotent group is the direct product of pgroups. The trivial group is a group of nilpotency class two in fact, it has class zero. Recall that if gis a group then zg fa2g jab bafor all b2gg. In 1, bacon and kappe give a classification of 2generator pgroups of nilpotency class 2 with the goal of computing their. Michaux state forest bureau of forestry michaux state forest 34 174 696 997 997 233 234 394 233 carlisle. The semidihedral group also denoted is the semidihedral group also called quasidihedral group of order. A group g of nilpotency class c may have a central series of length greater than c. Computing the nonabelian tensor square of metacyclic pgroups of nilpotency class two.
The coexponent of a finite pgroup is introduced and we consider how the nilpotency class is bounded in terms of this invariant. 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. Independent of the nilpotency class, this result is also true if we consider a powerful finite pgroup cf. We find the nilpotency class of a group of 2symmetric words for free. The nilpotency class of pgroups in which subgroups have.
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. The multiplicative group of upper unitriangular n x n matrices over any field f is a nilpotent group of nilpotency class n 1. 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. In particular, all 3groups with maximal nilpotency class and order at least 34 have 6 conjugacy classes of maximal size 2, 14. This is a p group of order smaller than g,soby induction it is nilpotent, say. A nilpotent group can be thought of a group that is only finitely removed from an abelian group. G is a normal sylow psubgroup of autg, when g is an extraspecial pgroup. 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. My guess is no, at least as regards a classification.
This is a pgroup of order smaller than g,soby induction it is nilpotent, say. The nilpotency class of pgroups in which subgroups have few conjugates giovanni cutolo, howard smith and james wiegold abstract. The qtensor square of a powerful pgroup sciencedirect. 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. Corollary 2 let g be a nite 2group with conjugate type 1. This is the first part of two papers devoted to compute the nilpotency class of. By the equivalence of definitions of finite nilpotent group, every finite nilpotent group is a direct product of its sylow subgroups. 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. 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. 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.
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. Part i concerns the following counting function and various refinements. 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. 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. Since groups of order p p are always regular groups, it is also a minimal such example. A bound for the nilpotency class of a finite pgroup in. If gis a nilpotent group then the nilpotency class of gis the smallest n 0 such that z ng g. Finite pgroups with the least number of outer pautomorphisms. Muller 16, theorem has shown that if g is neither elementary abelian nor extraspecial, then aut.
The group can also be defined as the general semilinear group of degree one over the field of nine elements arithmetic functions. In particular, we prove that every finite pgroup is nilpotent for a prime p. Nilpotency class of symmetric units of group algebras. W e note that a pgroup of order p n with a conjugacy class of length p n. A survey 4 in this part we consider pgroups of small orders and mention results on class preservingautomorphismsof pgroupsoforderat most p5. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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. Since zg is a nitely generated abelian group, our task. Pdf computing the nonabelian tensor square of metacyclic. We consider the problem of how the nilpotency class. 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. A pgroup of order p and nilpotency class c is said to have coclass rn c. A new algorithm for finding the nilpotency class of. The coclass graph associated with groups of maximal class.
Using transfinite recursion, the notion of nilpotency class can be extended to any ordinal. 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. A pgroup g of maximal nilpotency class and order at least p4 has p. We would like to show you a description here but the site wont allow us. Examples of groups of larger nilpotency class and satisfying hypothesis a are given below. The nilpotency class of pgroups in which subgroups have few. Nilpotency degree of integral cohomology classes of p. 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. Pdf conjugacy classes and characters of finite p groups. Burnside theorem and finite pnilpotent groups iazd. We sharpen some previously known bounds and also prove that a pgroup in which every cyclic subgroup has at most p2 conju gates has class. Also, a nonabelian group 2010 mathematics subject classi cation. For more information of the number of pgroups of various nilpotency class values for various primes, refer nilpotency class distribution of pgroups.
For instance, if g is abelian, then it has class 1. The 2groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups. The following result is a consequence of corollary 3. 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. The coexponent of a finite p group is introduced and we consider how the nilpotency class is bounded in terms of this invariant. In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal psubgroup.