On 2-symmetric words for groups

Abstract

A binary word wxY y is called 2-symmetric for a given group G if wgY h whY g for all gY h in G. In this note we describe 2-symmetric words for the relatively free (nilpotent of class 2)-by-abelian groups and for the relatively free centreby-metabelian groups.1. Introduction. Symmetric words for a group G are closely related to the fixed points of the automorphisms permuting generators in their corresponding relatively free groups. The problem of characterizing the 2-symmetric words for a given group G was initiated by Pøonka ([6], [7]) who, among other things, gave a complete description of the 2-symmetric words for nilpotent groups of class % 3. Further descriptions of 2-symmetric words are now known for the free metabelian groups and free soluble groups of derived length 3 (Macedon  ska and Solitar [4]) and free metabelian nilpotent groups of arbitrary nilpotency class c (Hoøubowski [2]). Further work for free nilpotent groups is under investigation by the second author ([3]). In this note we extend the work in [4] and characterize the 2-symmetric words for free centre-by-metabelian groups (Theorem 3) and for free (nilpotent of class 2)-by-abelian groups (Theorem 2).

Fixed points of automorphisms preserving the length of words in free solvable groups

“…In [15] P lonka described symmetric words in nilpotent groups of class ≤ 3, and it follows from his description that non-abelian nilpotent groups do not belong to K. In the series of papers [4-6] Ho lubowski described symmetric words in free nilpotent groups of class 4 and 5, 2-and 3-symmetric words in free metabelian groups and in free metabelian nilpotent groups of any class. In [3] Gupta and Ho lubowski found all 2-symmetric words in free nilpotentby-abelian groups and free centre-by-metabelian groups. In [10] Macedońska and Solitar characterized 2-symmetric words in free metabelian and solvable groups of derived length 3.…”

On symmetric words in the symmetric group of degree three

“…Thus the group S (2) (S 3 ) is the direct product of subgroup gp{s, t} ∼ = S 3 and cyclic group gp{v} of order 3.…”

“…It has been done for arbitrary nilpotent groups of class ≤ 3 and in the case of symmetric 2-words also for dihedral group of order 2p. In the papers [2] and [9] 2-words are determined for free metabelian groups and soluble groups of derived length 3. A description of the groups S (2) (G) and S (3) (G) for free metabelian and free metabelian, nilpotent group G is given in [4].…”

“…In the papers [2] and [9] 2-words are determined for free metabelian groups and soluble groups of derived length 3. A description of the groups S (2) (G) and S (3) (G) for free metabelian and free metabelian, nilpotent group G is given in [4]. The same for free nilpotent groups of class 4 and 5 has been done in [5], [6] and [7].…”

“…In this note all symmetric 2-words for the symmetric group S 3 are listed and using this we determine the group S (3) (S 3 ). Unexpectedly enough it turns out that it is isomorphic to the group (Z 3 ) 6 , whereas the group S (2) (S 3 ) is non-Abelian. Moreover we prove that all groups S (n) (S 3 ) with the exception n = 2 are commutative.…”