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Counterexamples to congruences in finite p-groups
Abstract
Let K be a conjugacy class of a finite p-group G. Héthelyi, Külshammer and the author asked whether the number of conjugacy classes contained in KK−1 is always congruent to $1$ modulo p−1, and whether, dually, the number of irreducible constituents of χχ̅ is congruent to $1$ modulo p−1 for every χ∈Irr(G). We construct a group of order 77 for which both congruences fail.
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BibTeX
@misc{sambale2026counterexamples,
title = {{Counterexamples to congruences in finite p-groups}},
author = {Benjamin Sambale},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33615},
eprint = {2610.33615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33615},
}
RIS
TY - UNPB
TI - Counterexamples to congruences in finite p-groups
AU - Benjamin Sambale
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33615
UR - https://arxiv.org/abs/2610.33615
AB - Let K be a conjugacy class of a finite p-group G. Héthelyi, Külshammer and the author asked whether the number of conjugacy classes contained in KK^−1 is always congruent to $1$ modulo p−1, and whether, dually, the number of irreducible constituents of χχ̅ is congruent to $1$ modulo p−1 for every χ∈Irr(G). We construct a group of order 7^7 for which both congruences fail.
ER -