104 research outputs found
On the number of conjugacy classes of a permutation group
We prove that any permutation group of degree has at most
conjugacy classes.Comment: 9 page
The minimal base size for a p-solvable linear group
Let be a finite vector space over a finite field of order and of
characteristic . Let be a -solvable completely reducible
linear group. Then there exists a base for on of size at most
unless in which case there exists a base of size at most . The
first statement extends a recent result of Halasi and Podoski and the second
statement generalizes a theorem of Seress. An extension of a theorem of P\'alfy
and Wolf is also given.Comment: 11 page
A lower bound for the number of conjugacy classes of a finite group
Every finite group whose order is divisible by a prime has at least conjugacy classes.Comment: references added to the Introductio
Normalizers of Primitive Permutation Groups
Let be a transitive normal subgroup of a permutation group of finite
degree . The factor group can be considered as a certain Galois group
and one would like to bound its size. One of the results of the paper is that
if is primitive unless , , , , or
. This bound is sharp when is prime. In fact, when is
primitive, unless is a member of a given infinite
sequence of primitive groups and is different from the previously listed
integers. Many other results of this flavor are established not only for
permutation groups but also for linear groups and Galois groups.Comment: 44 pages, grant numbers updated, referee's comments include
Finite groups have more conjugacy classes
We prove that for every there exists a so that
every group of order has at least conjugacy classes. This sharpens earlier results of
Pyber and Keller. Bertram speculates whether it is true that every finite group
of order has more than conjugacy classes. We answer Bertram's
question in the affirmative for groups with a trivial solvable radical
Set-Direct Factorizations of Groups
We consider factorizations where is a general group, and
are normal subsets of and any has a unique representation
with and . This definition coincides with the customary and
extensively studied definition of a direct product decomposition by subsets
of a finite abelian group. Our main result states that a group has such
a factorization if and only if is a central product of and and the central subgroup
satisfies
certain abelian factorization conditions. We analyze some special cases and
give examples. In particular, simple groups have no non-trivial set-direct
factorization
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