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.

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.

A well known theorem of G. A. Miller [4] (see also [2]) shows that a p-group of order p" where n > v(v 1)/2 contains an Abelian subgroup of order p° . It is clear that this theorem together with Sylow's Theorem implies that any finite group of large order contains an Abelian p-group of large order . In this note we use simple number theoretic considerations to make this implication more precise...

The Z64-algebra of the genetic code and DNA sequences of length N was recently stated. In order to beat the limits of this structure −such as the impossibility of non-coding region analysis in genomes and the impossibility of the insertions and deletions analysis (indel mutations)− we have develop a cycle group structure over the of extended base triplets of DNA X1X2X3, Xi∈{O, A, C, G, U}, wher...

A set of quasi-uniform random variables X1, . . . , Xn may be generated from a finite group G and n of its subgroups, with the corresponding entropic vector depending on the subgroup structure of G. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in mo...

Let Π be a finite projective plane admitting a large abelian collineation group. It is well known that this situation may be studied by algebraic means (via a representation by suitable types of difference sets), namely using group rings and algebraic number theory and leading to rather strong nonexistence results. What is less well-known is the fact that the abelian group (and sometimes its gr...

‎motivated by a celebrated theorem of schur‎, ‎we show that if~$gamma$ is a normal subgroup of the full automorphism group $aut(g)$ of a group $g$ such that $inn(g)$ is contained in $gamma$ and $aut(g)/gamma$ has no uncountable abelian subgroups of prime exponent‎, ‎then $[g,gamma]$ is finite‎, ‎provided that the subgroup consisting of all elements of $g$ fixed by $gamma$ has finite index‎. ‎so...