P. C. Wong

Residual finiteness of outer automorphism groups of certain tree products

Abstract We prove that the outer automorphism groups of tree products of finitely many polycyclic-by-finite groups amalgamating central edge groups are residually finite. As a consequence, the outer automorphism groups of tree products of finitely many abelian groups are residually finite.

Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

In this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.

On Engel Elements of a Group Relative to Certain Subgroup

An element g in a group G is called a left Engel element of G, if for each x ∈ G, there is a positive integer n = n(g, x) such that [x, n g] = 1. In this article, we will study a generalization of the left Engel elements and its connections with the generalized Hirsch–Plotkin and Baer radical.

POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

A group G is called cyclic subgroup separable for the cyclic subgroup H if for each x ∈ G\H, there exists a normal subgroup N of finite index in G such that x / ∈ HN . Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-byfinite groups.

ON CERTAIN PAIRS OF NON-ENGEL ELEMENTS IN FINITE GROUPS

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair if h = [h,n g] and g = [g,n h]. In this paper, we will study the subgroup generated by the n-Engel pair under certain conditions.

ON n-ENGEL PAIR SATISFYING CERTAIN CONDITIONS

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,n g], g = [g,n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh-2gh = ghg and ghg-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.

CYCLIC SUBGROUP SEPARABILITY OF CERTAIN GRAPH PRODUCTS OF SUBGROUP SEPARABLE GROUPS

In this paper, we show that tree products of certain subgroup separable groups amalgamating normal subgroups are cyclic subgroup separable. We then extend this result to certain graph product of certain subgroup separable groups amalgamating normal subgroups, that is we show that if the graph has exactly one cycle and the cycle is of length at least four, then the graph product is cyclic subgroup separable.

Weak potency of certain fundamental groups of graphs of groups

It is known that the fundamental group of graph of polycyclic-by-finite and free-by-finite groups amalgamating infinite cyclic edge subgroups with non-trivial intersection are weakly potent. In this paper, we shall show that these groups are still weakly potent when the amalgamated infinite cyclic edge subgroups have trivial intersection.

On the existence of Engel pairs in certain linear groups

Let G be a group and h,g ∈ G. The 2-tuple (h,g) is said to be an n-Engel pair, n ≥ 2, if h = [h,ng], g = [g,nh] and h≠1. Let SL(2,F) be the special linear group of degree 2 over the field F. In this paper, we show that given any field L, there is a field extension F of L with [F : L] ≤ 6 such that SL(2,F) has an n-Engel pair for some integer n ≥ 4. We will also show that SL(2,F) has a 5-Engel pair if F is a field of characteristic p ≡±1mod 5.

The Weakly Potency of Certain HNN Extensions of Nilpotent Groups

In this paper we give a characterization for certain HNN extensions of subgroup separable groups with normal associated subgroups to be weakly potent. We then apply our result to show that certain HNN extensions of finitely generated nilpotent groups with central associated subgroups are weakly potent.