# Impartial Games for Generating Groups

## by Dana Ernst

Published April 23, 2013 in Research

Loosely speaking, a group is a set together with an associative binary operation that satisfies a few modest conditions: the "product" of any two elements from the set is an element of the set (closure), there exists a "do nothing" element (identity), and for every element in the set, there exists another element in the set that "undoes" the original (inverses). Let G be a finite group. Given a single element from G, we can create new elements of the group by raising the element to various powers. Given two elements, we have even more options for creating new elements by combining powers of the two elements. Since G is finite, some finite number of elements will "generate" all of G. In the game DO GENERATE, two players alternately select elements from G. At each stage, a group is generated by the previously selected elements. The winner is the player that generates all of G. There is an alternate version of the game called DO NOT GENERATE in which the loser is the player that generates all of G. In this talk, we will explore both games and discuss winning strategies. Time permitting, we may also relay some current research related to both games.

This talk was given on April 23, 2013 as part of the Cool Math Talk series at the University of Nebraska at Omaha.