We describe all the fine group gradings, up to equivalence, on the Lie algebra d4. This problem is equivalent to finding the maximal abelian diagonalizable subgroups of the automorphism group of d4. We prove that there are fourteen by using two different viewpoints. The first approach is computational: we get a full description of the gradings by using a particular implementation of the automor...

A $p$-group $G$ is called a $mathcal{CAC}$-$p$-group if $C_G(H)/H$ is ‎cyclic for every non-cyclic abelian subgroup $H$ in $G$ with $Hnleq‎ ‎Z(G)$‎. ‎In this paper‎, ‎we give a complete classification of‎ ‎finite $mathcal{CAC}$-$p$-groups‎.

The Non-Abelian finite group PSL2(7) is the only simple subgroup of SU(3) with a complex three-dimensional irreducible representation. It has two maximal subgroups, S4 which, along with its own A4 subgroup, has been successfully applied in numerous models of flavor, as well as the 21 element Frobenius group Z7⋊Z3, which has gained much less attention. We show that it can also be used to generat...

The Non-Abelian finite group PSL2(7) is the only simple subgroup of SU(3) with a complex three-dimensional irreducible representation. It has two maximal subgroups, S4 which, along with its own A4 subgroup, has been successfully applied in numerous models of flavor, as well as the 21 element Frobenius group Z7⋊Z3, which has gained much less attention. We show that it can also be used to generat...

Abstract In this article, some new sufficient conditions of p -nilpotency finite groups are obtained by using c -normality and Φ-supplementary the maximal or 2-maximal subgroups Sylow -subgroups.

‎let $v$ be a vector space over a field $f$ of characteristic $pgeq 0$ and let‎ ‎$t$ be a regular subgroup of the affine group $agl(v)$‎. ‎in the finite dimensional case we show that‎, ‎if $t$ is abelian or $p>0$‎, ‎then $t$ is unipotent‎. ‎for $t$ abelian‎, ‎pushing forward some ideas used in [a‎. ‎caranti‎, ‎f‎. ‎dalla volta and m‎. ‎sala‎, ‎abelian regular subgroups of the affine group and r...

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...

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class o...