In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...

In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the s...

We determine the upper central series of the maximal normal p–subgroup of the automorphism group of a bounded abelian p–group.

it is shown that there are exactly seventy-eight 3-generator 2-groups of order $2^{11}$ with trivial schur multiplier. we then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero.