2016/04/05 by Mario Gómez, Gómez, Mario, Sergio R. López-Permouth +7
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1604.01119
openalex publication_date 2016/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A binary operation on any set induces a binary operation on its subsets. We explore families of subsets of a group that become a group under the induced operation and refer to such families as power groups of the given group. Our results serve to characterize some types of groups in terms of their power groups. In particular, we consider when the only power groups of a group are the factor groups of its subgroups and when that is the case up to isomorphism. We prove that the former are precisely those groups for which every element has finite order and provide examples to illustrate that the latter is not always the case. In the process we consider several natural questions such as whether the identity element of the group must belong to the identity element of a power group or the inverse of an element in a power group must consist of the inverses of its elements.