This paper, deals with some equivalence relations in fuzzy subgroups. Further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

This review is intended to provide the reader who is interested in the historical development of the recent research on photoremovable protecting groups, phototriggers, and “caged” compounds and to provide some insight into the genesis of our entry into the field through our most recent discoveries. Several other investigations have paralleled our development of a photoremovable protecting grou...

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/X(S...

Consider the question: which finite groups operate as symmetries of the complex projective plane P’(C)? Any finite subgroup of &X,(C) acts as a group of collineations and these give the linear models. The list of such groups is relatively short [MBD] but contains, for example, a groups of rank < 2, subgroups of U(Z), and the simple groups A,, A,, an PSL(F,). It turns out that these linear group...

Compact Lie groups appear frequently in algebraic topology, but they are relatively rigid analytic objects, and partially for that reason are a challenge to understand homotopically. Since H. Hopf (and H-spaces) topologists have aspired to escape the analytic straitjacket by finding some homotopy theoretic concept which would capture enough of the idea of “compact Lie group” to lead to rich and...

We show that if an inclusion of finite groups H ≤ G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F–isomorphism in mod p cohomology, then H controls p–fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p–subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology i...