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Toward a classification of the supercharacter theories of Cp × Cp
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 4 / November 2022
- Published online by Cambridge University Press:
- 12 October 2022, pp. 990-1010
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- Article
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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 $p$
.
