Relator Games on Groups
Abstract
We define two impartial games, the Relator Achievement Game REL and the Relator Avoidance Game RAV. Given a finite group G and generating set S, both games begin with the empty word. Two players form a word in S by alternately appending an element from S S-1 at each turn. The first player to form a word equivalent in G to a previous word wins the game REL but loses the game RAV. Alternatively, one can think of REL and RAV as make a cycle and avoid a cycle games on the Cayley graph (G,S). We determine winning strategies for several families of finite groups including dihedral, dicyclic, and products of cyclic groups.
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