Toward a classification of the supercharacter theories of Cp × Cp. Issue 4 (12th November 2022)

• Record Type: Journal Article
• Authors: Burkett, Shawn T.; Lewis, Mark L.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 65:Issue 4(2022)
• Page Start: 990
• Page End: 1010
• Publication Date: 2022-11-12
• Subjects: supercharacter theories -- elementary abelian p-groups -- automorphisms; 20C15 -- 20D15; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091522000438
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: In this paper, we study the supercharacter theories of elementary abelian $p$ -groups of order $p^{2}$ . We show that the supercharacter theories that arise from the direct product construction and the $\ast$ -product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$ -group of order $p^{2}$ that has a non-identity superclass of size $1$ or a non-principal linear supercharacter must come from either a $\ast$ -product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2, \, 3, \, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$ .