Date of Graduation

Summer 2012

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Les Reid

Abstract

In this thesis, we investigate the set of powers in a group G. We study Fr_n (G), the fraction of elements of G that are n-th powers, and under what conditions the set of n-th powers, G^((n) ) is a subgroup of G. Given a group G as a product of cyclic groups of prime power order, we are able to determine Fr_n (G). We also determine Fr_n for dihedral groups and provide methods to determine Fr_n of both symmetric and alternating groups. We show that we can construct G such that Fr_2 (G)=1/n for any positive integer n. We establish that Fr_2 (G) can be arbitrarily close to 1, and that 〖{Fr〗_2 (G)} is dense in the interval [0,1]. Conditions involving the nilpotency of G are determined that guarantee that the set of n-th powers of G will be a subgroup of G. Finally, we conjecture that Fr_2 (G) can be any rational value in (0,1].

Keywords

group theory, set of squares, set of cubes, set of powers, semidirect products, commutators, nilpotency

Subject Categories

Mathematics

Copyright

© Caleb A. Bennett

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