Large orbits of p-groups on conjugacy classes
Abstract
Let P be a p-group that acts faithfully and coprimely on a finite group H. A classical result of D.\ S.\ Passman, in the form proved by I.\ M.\ Isaacs, asserts that there exists an element x∈ H such that |CP(x)|≤ (|P|/p)1/p. In this paper, we study the corresponding question for the action of P on the conjugacy classes of H. We show that for every odd prime p the analogue of Isaacs' bound fails for classes. On the other hand, we prove that if H is solvable, then there always exists a conjugacy class c of H whose stabilizer in P has order at most |P|1/2, and that this bound is best possible for every prime p. As a consequence, there always exists an irreducible character of H whose inertia group in P has order at most |P|1/2. The proof proceeds via a result on linear actions of p-nilpotent groups, and in the quasiprimitive case it relies on a classification of Holt and Yang and on computer calculations.
Related
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BibTeX
@misc{keller2026large,
title = {{Large orbits of $p$-groups on conjugacy classes}},
author = {Thomas Michael Keller},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05904},
eprint = {2611.05904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05904},
}
RIS
TY - UNPB
TI - Large orbits of p-groups on conjugacy classes
AU - Thomas Michael Keller
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05904
UR - https://arxiv.org/abs/2611.05904
AB - Let P be a p-group that acts faithfully and coprimely on a finite group H. A classical result of D.\ S.\ Passman, in the form proved by I.\ M.\ Isaacs, asserts that there exists an element x∈ H such that |C_P(x)|≤ (|P|/p)^1/p. In this paper, we study the corresponding question for the action of P on the conjugacy classes of H. We show that for every odd prime p the analogue of Isaacs' bound fails for classes. On the other hand, we prove that if H is solvable, then there always exists a conjugacy class c of H whose stabilizer in P has order at most |P|^1/2, and that this bound is best possible for every prime p. As a consequence, there always exists an irreducible character of H whose inertia group in P has order at most |P|^1/2. The proof proceeds via a result on linear actions of p-nilpotent groups, and in the quasiprimitive case it relies on a classification of Holt and Yang and on computer calculations.
ER -