Bret J. Benesh

The Metric Metric on S4

My favorite band is currentlyMetric.My kids’ favorite band is currentlyMetric. Our favorite video is “Gimme Sympathy” [1], in which the band members switch instruments several times. Below, we translate their movement among the different instruments into the language of permutations. We can then use these permutations to define a distance function, or metric, on the set of all permutations of four objects, known as the symmetric group of degree 4 (denoted S4). To do this, we employ the word metric, which is an idea from the field of geometric group theory (see [3], for example). In the end, we define the deliciously namedMetric metric on S4.

A q-player impartial avoidance game for generating finite groups

We study a q-player variation of the impartial avoidance game introduced by Anderson and Harary, where q is a prime. The game is played by the q players taking turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning. We introduce a ranking system for the other players to prevent coalitions. We describe the winning strategy for these games on cyclic, nilpotent, dihedral, and dicyclic groups.