z-classes in groups: a survey

This survey article explores the notion of z-classes in groups. The concept introduced here is related to the notion of orbit types in transformation groups, and types or genus in the representation theory of finite groups of Lie type. Two elements in a group are said to be z-equivalent (or z-conjugate) if their centralizers are conjugate. This is a weaker notion than the conjugacy of elements. In this survey article, we present several known results on this topic and suggest some further questions

Groups with star free commuting graphs

Let $G$ be a group and $Z(G)$ be its center. We associate a commuting graph ${\Gamma}(G)$, whose vertex set is $G\setminus Z(G)$ and two distinct vertices are adjacent if they commute. We say that ${\Gamma}(G)$ is strong $k$ star free if the $k$ star graph is not a subgraph of ${\Gamma}(G)$. In this paper, we characterize all strong $5$ star free commuting graphs. As a byproduct, we classify all strong claw-free graphs. Also, we prove that the set of all non-abelian groups whose commuting graph is strong $k$ star free is finite.Comment: 12 pages, 4 figure

The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm for Symplectic and Orthogonal Groups

In this chapter, we study the MOR cryptosystem with symplectic and orthogonal groups over finite fields of odd characteristics. There are four infinite families of finite classical Chevalley groups. These are special linear groups SL(d, q), orthogonal groups O(d, q), and symplectic groups Sp(d, q). The family O(d, q) splits into two different families of Chevalley groups depending on the parity of d. The MOR cryptosystem over SL(d, q) was studied by the second author. In that case, the hardness of the MOR cryptosystem was found to be equivalent to the discrete logarithm problem in F q d . In this chapter, we show that the MOR cryptosystem over Sp(d, q) has the security of the discrete logarithm problem in F q d . However, it seems likely that the security of the MOR cryptosystem for the family of orthogonal groups is F q d 2 . We also develop an analog of row-column operations in symplectic and orthogonal groups which is of independent interest as an appendix