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Abstract
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The number
of
-regular classes
of a finite group
is a key invariant in modular representation theory. Several outstanding conjectures propose
that this number can be calculated or bounded in terms of certain invariants of some subgroups
of
. Our main
question here is if
can be bounded by the number of conjugacy classes of some subgroup of
of order not
divisible by
.
This would have consequences for the Malle–Robinson
-conjecture. Furthermore,
we investigate a
-version
of this, for sets of primes
.
As part of our investigations, we study finite groups that have more conjugacy classes than any of their
proper subgroups. These groups naturally appear in questions on bounding from above the number of
conjugacy classes of a group, and were considered by G. R. Robinson and J. G. Thompson in the context
of the
-problem. We classify
the almost Abelian groups
with
quasisimple. Our results should be of use in several related questions.
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Keywords
number of conjugacy classes, $\pi$-subgroups, almost
Abelian finite groups, $l(B)$-conjecture
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Mathematical Subject Classification
Primary: 20C20, 20D20, 20D60
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Milestones
Received: 26 May 2020
Revised: 8 January 2021
Accepted: 12 January 2021
Published: 17 March 2021
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