Groups of Prime Power Order with Many Conjugacy Classes (original) (raw)

Finite p-central groups of height k

2011

A finite group G is called p i-central of height k if every element of order p i of G is contained in the k th-term ζ k (G) of the ascending central series of G. If p is odd such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent pgroups, i.e., for such a finite p-group P the group P/Ω 1 (P) is also p-central of height p − 2 (Thm. B). In such a group P the index of P p is less or equal than the order of the subgroup Ω 1 (P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl p (G) is p-central of height p − 2, then N G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).

Nilpotency: A Characterization Of The Finite p-Groups

Journal of Mathematics , 2017

Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]

Finite p-Groups of Nilpotency Class 3 with Two Conjugacy Class Sizes

Israel Journal of Mathematics

It is proved that, for a prime p > 2 and an integer n ≥ 1, finite p-groups of nilpotency class 3 and having only two conjugacy class sizes 1 and p n exist if and only if n is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).

Finite p -groups having Schur multiplier of maximum order

Journal of Algebra

Let G be a non-abelian p-group of order p n and M (G) denote the Schur multiplier of G. Niroomand proved that |M (G)| ≤ p 1 2 (n+k−2)(n−k−1)+1 for non-abelian p-groups G of order p n with derived subgroup of order p k. Recently Rai classified p-groups G of nilpotency class 2 for which |M (G)| attains this bound. In this article we show that there is no finite p-group G of nilpotency class c ≥ 3 for p = 3 such that |M (G)| attains this bound. Hence |M (G)| ≤ p 1 2 (n+k−2)(n−k−1) for p-groups G of class c ≥ 3 where p = 3. We also construct a p-group G for p = 3 such that |M (G)| attains the Niroomand's bound. 2 (n+k−2)(n−k−1)+1 we shall write |M (G)| attains the bound, throughout this paper. Recently Rai [16, Theorem 2.1] classified finite p-groups G of class 2 such that |M (G)| attains the bound. Aim of this paper is to continue this line of investigation and to look into the classification of arbitrary finite p-groups attending this bound. It, surprisingly turns out that for p = 3 there is no finite p-group G of nilpotency class c ≥ 3 such that |M (G)| attains the bound. Hence for p-groups G of class ≥ 3 and p = 3 we improve 2010 Mathematics Subject Classification. 20J99, 20D15.

Powerful p-groups. I. Finite groups

Journal of Algebra, 1987

In this paper we study a special class of finite p-groups, which we call powerful p-groups. In the second part of this paper, we apply our results to the study of p-adic analytic groups. This application is possible, because a finitely generated prop group is p-adic analytic if and only if it is "virtually pro-powerful." These applications are described in the introduction to the second part, while now we describe the present part in more detail. In the first section we define a powerful p-group, as one whose subgroup of pth powers contains the commutator subgroup. We give several results

A note on p-groups of order ≤ p 4

Proceedings - Mathematical Sciences, 2009

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p 4 , then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.