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A self-normalizing subgroup is always self-centralizing, but the converse is not necessarily true. Given a finite group G, we denote by w(G) the number of all selfcentralizing subgroups of G which are not self-normalizing. We observe that w(G) = 0 if and only if G is abelian, and that if G is nonabelian nilpotent then w(G) ≥ 3. We also prove that if w(G) ≤ 20 then G is solvable. Finally, we provide structural information in the case when w(G) ≤ 3. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a group is nilpotent of class at most two if and only if each of its self-centralizing subgroups is normal. We also show that if all self-centralizing subgroups of a certain group are subnormal then all subgroups are subnormal. [ABSTRACT FROM AUTHOR]

In this paper, we propose a more secure version of a Cayley hash function which is based on the linear functions. It is a practical parallelizable hash function. [ABSTRACT FROM AUTHOR]

The Paul Erdős and András Gyárfás conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order $$p^3$$ . [ABSTRACT FROM AUTHOR]

In this paper, we extend upon the results of B. Suceavă and R. Stong [Amer. Math. Monthly, 110 (2003) 162{162], which they computed the minimum number of 3-cycles needed to generate an even permutation. Let Ωnk,m be the set of all permutations of the form c1c2 · · · ck where ci's are arbitrary m-cycles in Sn. Suppose that Γnk,m be the Cayley graph on subgroup of Sn generated by all permutations in Ωnk,m. We find a shortest path joining identity and any vertex of Γnk,m for arbitrary natural number k, and m = 2, 3, 4. Also, we calculate the diameter of these Cayley graphs. As an application, we present an algorithm for finding a short expression of a permutation as products of given permutations. [ABSTRACT FROM AUTHOR]

In this paper, we classify the finite non-solvable groups satisfying the following property P: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes. [ABSTRACT FROM AUTHOR]