Palindromes in Finite Groups

Abstract

In this work, we begin by giving an overview of some topics in group theory, namely semidirect products, nilpotent groups and wreath products. We use wreath products to prove Schur's theorem which says that if the order and index of a normal subgroup A of a group G are relatively prime, then the A has a complement in G. Next, we introduce the notion of a civic group: a group with the property that every subset which is closed under taking palindromes is a subgroup. We prove that civic groups satisfy the property that its palindromic width is equal to one and then we reduce the classification of civic groups to the odd order case. More precisely, we show that every civic group is a direct product of a cyclic 2-group and a civic group of odd order. Further, we show that a minimal group of odd order having palindromic width greater than 1 is a semidirect product of two elementary abelian groups, or a p-group. This is also the form of the minimal non-civic groups of odd order. Finally, we show that for solving the so-called Magnus-Derek game on general finite groups, it suffices to consider the odd order case. We give a solution of the game for civic groups of odd order, as well as other groups having sufficiently large subgroups. Moreover, we make progress on the solution of the game in general groups by giving a solution in terms of a maximal subset closed under taking palindromes. We conjecture that such subsets can in fact always be chosen to be subgroups, but that question remains open for now.