Date of Graduation

Summer 2012

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Les Reid

Keywords

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

Subject Categories

Mathematics

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].

Copyright

© Caleb A. Bennett

Campus Only

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