For a group G, embedded in its of permutations B=Perm(G) via the left regular representation λ:G→B, normalizer λ(G) B is Hol(G), holomorph G. The set ℋ(G) those N≤Hol(G) such that N≅G and NormB(N)=Hol(G) keyed to structure so-called multiple NHol(G)=NormB(Hol(G)), conjugates by NHol(G). We wish generalize this considering certain

This paper is expository. It is mainly a survey of some of our work on the algebraic systems that arise in fuzzy set theory and logic. We include some of the proofs here and refer to the papers [18, 19, 20, 21, 22] for further details. Our point of view is the algebraic one: when are the various systems that arise isomorphic, and what are their symmetries (automorphisms)? The bulk of the materi...

We consider normalizers of an irreducible inclusion N ⊆ M of II1 factors. In the infinite index setting an inclusion uNu∗ ⊆ N can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N ′ ∩ 〈M, eN 〉 is of the form ueNu for ...

1. The universal p-Frattini cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The p-Frattini module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Restriction to the normalizer of a p-Sylow. . . . . . . . . . . . . . . . . . . . . . . 8 4. Asymptotics of the p-Frattini modules Mn . . . . . . . . . . . . . . . . . . . . ....

Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...