Normal subgroups of powerful p -groups
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NORMAL SUBGROUPS OF POWERFUL p-GROUPS
BY
James Williams School of Mathematics, University of Bristol Fry Building, Woodland Road, Bristol BS8 1UG, UK e-mail: [email protected]
ABSTRACT
In this note we show that if p is an odd prime and G is a powerful p-group with N ≤ Gp and N normal in G, then N is powerfully nilpotent. An analogous result is proved for p = 2 when N ≤ G4 .
1. Introduction Powerfully nilpotent groups, introduced in [8], are a type of powerful p-group, possessing a central series of a special kind. This family of groups has a rich and beautiful structure theory. For example, to each powerfully nilpotent group we can associate a quantity known as the powerful coclass, and it turns out that the rank and exponent of a powerfully nilpotent group can be bounded in terms of their powerful coclass. Many characteristic subgroups of powerful p-groups are in fact powerfully nilpotent. In [8] it is shown that if G is powerful, then the proper terms of the derived and lower central series of G are powerfully i i nilpotent, as well as Gp for i ≥ 1. In [9] the Omega subgroups of Gp are shown to be powerfully nilpotent. In this note we prove the following for odd primes p: Theorem: For an odd prime p, if G is a powerful p-group and N ≤ Gp with N normal in G, then N is powerfully nilpotent. Received August 19, 2019
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J. WILLIAMS
Isr. J. Math.
For p = 2 we prove: Theorem: If G is a powerful 2-group and N ≤ G4 with N normal in G, then N is powerfully nilpotent. This builds on the results in [3] , which show that such a group N must be powerful and moreover provides an alternative proof for this fact. We also note that in [6], Mann poses the question: “Which p-groups are subgroups of powerful p-groups?”, and our result provides a partial answer in this direction.
2. Preliminaries In what follows all groups considered will be finite p-groups. First we shall recall some definitions and properties which will be used in the main part of this paper, with the aim of making this paper as self-contained as possible. Recall that H n = xn |x ∈ H is the subgroup generated by nth powers of elements of H. Definition: A subgroup N of a finite p-group G is powerfully embedded in G if [N, G] ≤ N p for p an odd prime, or [N, G] ≤ N 4 if p = 2. Definition: A finite p-group G is powerful if [G, G] ≤ Gp for p an odd prime, or [G, G] ≤ G4 if p = 2. Powerful p-groups were introduced in [5]. We will often make use of the following well known properties of powerful p-groups without explicit mention. Theorem 1: Let G be a powerful p-group, then: k
(1) [5, Proposition 1.7] For every k ∈ N the subgroup Gp coincides with k the set {xp |x ∈ G} of pk th powers of elements of G. k (2) [5, Corollary 1.2] Gp is powerfully embedded in G for all k ∈ N. (3) [5, Corollary 1.9] Suppose that G = a1 , . . . , ar is generated by elements a1 , . . . , ar . Then Gp = ap1 , . . . , apr . We also make use of the following result Lemma 2 ([7, Lemma 3.1]): If M and N are powerfully embedded subgroups i j i+j for all i, j ∈ N. in a finite
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