# Palindromes in finite groups and the Explorer-Director game

## TL;DR

This paper characterizes optimal strategies in the Explorer-Director game using twisted subgroups in finite groups, providing a complete solution for nilpotent groups and advancing understanding of group-based combinatorial games.

## Contribution

It introduces a structural approach using twisted subgroups to analyze the game, offering the first such characterization and solving it for all nilpotent groups.

## Key findings

- Reduced the game to finding the largest proper twisted subgroup.
- Provided a complete solution for the game on all nilpotent groups.
- Established a new connection between group theory and combinatorial game analysis.

## Abstract

In this paper, we use the notion of twisted subgroups (i.e., subsets of group elements closed under the binary operation $(a,b) \mapsto aba$) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus-Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.