On locally finite 𝑝-groups satisfying an Engel condition

Abstract

Abstract. For a given positive integer n and a given prime number p, let r = r(n, p) be the integer satisfying p r−1 < n ≤ p r . We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either p r or p r−1 if p is odd. When p = 2 the best upper bound is p r−1 , p r or p r+1 . In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.

On finite p-groups satisfying given laws

“…The details are given in Section 2. In particular, Theorem 1.4 generalizes a result by Abdollahi and Traustason [1] stating that the nilpotency class of n-Engel powerful p-groups can be bounded in terms of n only. A consequence of the above theorem is also the fact that AA ⊆ V if and only if there is a constant c such that for every prime p, every powerful pro-p group in V is nilpotent of class ≤ c.…”

“…Note that every finite metabelian p-group of coclass 1 belongs to AA p , see [6, p. 52]. This shows that (2) implies (1).…”

Quasi-powerful $p$-groups

“…is central, and hence abelian. Furthermore it is generated by the commutators of weight 2 in the generators of G; by Theorem 6.2 there are 1 2 r(r − 1) of these. Hence…”

“…For applications of powerful p-groups to the study of automorphisms of finite p-groups, see the excellent book [11]. For an application of powerful p-groups to the study of Engel groups, see [1]. Powerful p-groups have applications beyond finite p-groups, for instance they also have found uses in the study of p-adic analytic groups (see [13] and [3]).…”

Two Generator 4-Engel Groups

“…Every 4-Engel group that is locally nilpotent and without elements of order 2, 3 or 5 is nilpotent of class at most 7 [12]. If only the primes 2 and 5 are excluded the groups are not in general nilpotent but they are still soluble [1]. On the other hand there are examples of locally nilpotent 4-Engel 2-groups and 5-groups that are not soluble [2,11].…”