Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

If a non-trivial subgroup A of the group of continuous automorphisms of a noncyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F>!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension o...

In this paper, the notion of closure operators of matroids  is generalized to fuzzy setting  which is called $M$-fuzzifying closure operators, and some properties of $M$-fuzzifying closure operators are discussed. The $M$-fuzzifying matroid induced by an $M$-fuzzifying closure operator can induce an $M$-fuzzifying closure operator. Finally, the characterizations of $M$-fuzzifying acyclic matroi...

*Renier J. Brentjens,1-3 *Isabelle Rivière,1-4 Jae H. Park,1,2 Marco L. Davila,1,2 Xiuyan Wang,2-4 Jolanta Stefanski,2-4 Clare Taylor,2-4 Raymond Yeh,1,2 Shirley Bartido,2,3 Oriana Borquez-Ojeda,2-4 Malgorzata Olszewska,2-4 Yvette Bernal,1 Hollie Pegram,1,2 Mark Przybylowski,2-4 Daniel Hollyman,2-4 Yelena Usachenko,1,2 Domenick Pirraglia,2-4 James Hosey,2-4 Elmer Santos,3,5 Elizabeth Halton,1 P...

The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T.Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E....

Let Γg,b denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus g with b punctures. For a group G let Φf (G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φf (G) is nilpotent when G is a finitely generated subgroup of Γg,b, in this paper we compute Φf (G) for certain subgroups of Γg,b...