On Abelian Hall π-Subgroups

On non-normal non-abelian subgroups of finite groups

‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $Gcong{rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we s...

On N-high Subgroups of Abelian Groups

On Normal Abelian Subgroups in Parabolic Groups

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical Pu, and A a closed connected unipotent subgroup of Pu which is normalized by P. We show that P acts on A with nitely many orbits provided A is abelian. This generalizes a well-known niteness result, namely the case when A is central in Pu. We also obtain an analogous result for the adjoint action of P on in...

Extending Characters from Hall Subgroups

Suppose that G is a finite π-separable group. A classical result asserts that all irreducible characters of a Hall π-subgroup H of G extend to G if and only if H has a normal complement in G. Now, we fix a prime p and analyze when only the p-degree irreducible characters of H extend to G. 2010 Mathematics Subject Classification: 20C15 (primary), 20G40 (secondary)

COUNTING DISTINCT FUZZY SUBGROUPS OF SOME RANK-3 ABELIAN GROUPS

In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...