Quek, Shio Gai (2015) *Engel conditions in certain groups / Quek Shio Gai.* PhD thesis, University of Malaya.

## Abstract

This thesis is a study of certain Engel conditions. First, we will define the set of all the X-relative left Engel elements L(G;X) and the set of all the bounded X-relative left Engel elements L(G;X), where X is a subset of G. When X = G, L(G;X) = L(G) and L(G;X) = L(G), where L(G) is the set of all the usual left Engel elements and L(G) is the set of all the usual bounded left Engel elements. Next, we de�ne the X-relative Hirsch-Plotkin radical HP(G;X) and the X-relative Baer radical B(G;X). When X = G, HP(G;X) = HP(G) and B(G;X) = B(G) where HP(G) is the usual Hirsch-Plotkin radical and B(G) is the usual Baer radical. We will show that if X is a normal solvable subgroup of G, then B(G;X) = L(G;X) and HP(G;X) = L(G;X). This is an extension of the classical results B(G) = L(G) and HP(G) = L(G) provided that G is solvable. Next, we show that if X is a normal subgroup of G and G satis�es the maximal condition, then L(G;X) = HP(G;X) = B(G;X) = L(G;X), which is also an extension of the classical result L(G) = HP(G) = B(G) = L(G). We also proved similar results when X is a subgroup of certain linear groups. Let G be a group and h; g 2 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 6= 1. We will show that the subgroup generated by the 5-Engel pair (x; y) satisfying yxy = xyx and x5 = 1 is the alternating group A5. Next, we show that if (x; y) is an n-Engel pair, xyx

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