#### Thesis Title

#### 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

#### Recommended Citation

Bennett, Caleb A., "The Set of Powers in a Group" (2012). *MSU Graduate Theses*. 1643.

https://bearworks.missouristate.edu/theses/1643