584 research outputs found
Bounds for the relative n-th nilpotency degree in compact groups
The line of investigation of the present paper goes back to a classical work
of W. H. Gustafson of the 1973, in which it is described the probability that
two randomly chosen group elements commute. In the same work, he gave some
bounds for this kind of probability, providing information on the group
structure. We have recently obtained some generalizations of his results for
finite groups. Here we improve them in the context of the compact groups.Comment: 9 pages; to appear in Asian-European Journal of Mathematics with
several improvement
The probability that and commute in a compact group
We show that a compact group has finite conjugacy classes, i.e., is an
FC-group if and only if its center is open if and only if its commutator
subgroup is finite. Let denote the Haar measure of the set of all
pairs in for which ; this, formally, is the
probability that two randomly picked elements commute. We prove that is
always rational and that it is positive if and only if is an extension of
an FC-group by a finite group. This entails that is abelian by finite. The
proofs involve measure theory, transformation groups, Lie theory of arbitrary
compact groups, and representation theory of compact groups. Examples and
references to the history of the discussion are given at the end of the paper.Comment: 17 pages; we have cut some points ; to appear in Math. Proc.
Cambridge Phil. So
Relative n-isoclinism classes and relative n-th nilpotency degree of finite groups
The purpose of the present paper is to consider the notion of isoclinism
between two finite groups and its generalization to n-isoclinism, introduced by
J. C. Bioch in 1976. A weaker form of n-isoclinism, called relative
n-isoclinism, will be discussed. This will allow us to improve some classical
results in literature. We will point out the connections between a relative
n-isoclinism and the notions of commutativity degree, n-th nilpotency degree
and relative n-th nilpotency degree, which arouse interest in the
classification of groups of prime power order in the last years.Comment: 11 pages, to appear in Filomat with revision
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