Let [Formula: see text] be a group. Two elements [Formula: see text] are said to be in the same [Formula: see text]-class if their centralizers in [Formula: see text] are conjugate within [Formula: see text]. In this paper, we prove that the number of [Formula: see text]-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least [Formula: see text], and finite otherwise.

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.

Let [Formula: see text] be a group and [Formula: see text] be an automorphism of [Formula: see text]. Two elements [Formula: see text] are said to be [Formula: see text]-twisted conjugate if [Formula: see text] for some [Formula: see text]. We say that a group [Formula: see text] has the [Formula: see text]-property if the number of [Formula: see text]-twisted conjugacy classes is infinite for every automorphism [Formula: see text] of [Formula: see text]. In this paper, we prove that twisted Chevalley groups over a field [Formula: see text] of characteristic zero have the [Formula: see text]-property as well as the [Formula: see text]-property if [Formula: see text] has finite transcendence degree over [Formula: see text] or [Formula: see text] is periodic.

Let G be a group. An element g ∈ G is called reversible if it is conjugate to g −1 within G, and called strongly reversible if it is conjugate to g −1 by an order two element of G. Let H n H be the n-dimensional quaternionic hyperbolic space. Let PSp(n, 1) be the isometry group of H n H . In this paper, we classify reversible and strongly reversible elements in Sp(n) and Sp(n, 1). Also, we prove that all the elements of PSp(n, 1) are strongly reversible.

No abstract