Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c ∈ C and d ∈ D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an application of the famous marriage theorem of Philip Hall.
The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.
It is known that the centralizer of a matrix
over a finite field depends, up to conjugacy, only on the type of the
matrix, in the sense defined by J. A. Green.
In this paper an analogue of the type invariant is defined that
in general captures more information; using this invariant the
result on centralizers is extended to arbitrary fields.
The converse is also proved: thus two matrices have conjugate centralizers
if and only if they have the same generalized type.
The paper ends with the analogous results for symmetric and alternating groups.
For , let be the set of partitions of Ω given by the cycles of elements of G. Under the refinement order, admits join and meet operations.
We say that G is join- or meet-coherent if is join- or meet-closed, respectively.
The centralizer in of any
permutation g is shown to be meet-coherent, and join-coherent subject to a finiteness condition.
Hence if G is a centralizer in Sn, then is a lattice.
We prove that wreath products, acting imprimitively,
inherit join-coherence from their factors.
In particular automorphism groups of locally finite, spherically homogeneous trees are
We classify primitive join-coherent groups of finite degree, and also join-coherent subgroups of Sn
normalizing an n-cycle. We show that if is a chain, then there is a prime p such that G acts regularly on each of its orbits
as a subgroup of the Prüfer p-group, with G being isomorphic to an inverse limit of these subgroups.
This is the third in a series of papers whose object is to show how cycle index methods for finite classical groups, developed by Fulman [Jason Fulman. Cycle indices for the classical groups. J. Group Theory2 (1999), 251–289.], may be extended to other almost simple groups of classical type. In [John R. Britnell. Cyclic, separable and semisimple transformations in the special unitary groups over a finite field. J. Group Theory9 (2006), 547–569.] we treated the special unitary groups, and in [John R. Britnell. Cyclic, separable and semisimple transformations in the finite conformal groups. J. Group Theory9 (2006), 571–601.] the general symplectic and general orthogonal groups. In this paper we shall treat various subgroups of the general orthogonal group over a field of odd characteristic. We shall focus at first on Ω± (d, q), the commutator subgroup of Ο±(d, q). Subsequently we shall look at groups G in the range
Let G be a finite group and c be an element of . A subgroup H of G is said to be c-nilpotent
if it is nilpotent and has nilpotency class at most c. A subset X of G is said to be
non-c-nilpotent if it contains no two elements x and y such that the subgroup
is c-nilpotent. In this paper we study the quantity , defined to be the size of the largest non-c-nilpotent subset of L.
In the case that L is a finite group of Lie type, we identify covers of L by c-nilpotent subgroups, and
we use these covers to construct large non-c-nilpotent sets in the group L. We prove that for groups L of fixed rank r, there exist constants
Dr and Er such that , where N is the number of maximal tori in L.
In the case of groups L with twisted rank 1, we provide exact formulae for for all . If we write q for the level of the Frobenius endomorphism associated with L and assume that q > 5, then may be expressed as a polynomial in q with coefficients in .