On the existence of free symmetric pairs in normal subgroups of division rings with involution - Oliveira, Pedro Russo de (2020)
Let D be a noncommutative division ring with center k, whose characteristic is distinct from 2, endowed with an involution * -- which is said to be of the first kind, if it is k-linear, and of the second kind, otherwise. By a free symmetric pair in D, one understands a subset {x,y} of symmetric -- i.e., x* = x and y* = y -- nonzero members of D which freely generate a free group. Let N be a non central normal subgroup of the multiplicative group of D. We present sufficient conditions for the existence of free symmetric pairs in N, with exception of the case in which D is a quaternion algebra and * is symplectic. Specifically, when the dimension of D over k is finite, we show that N contains free symmetric pairs in the following cases: (a) * is of the first kind and k is uncountable; (b) D is a quaternion algebra and * is an orthogonal involution or an involution of the second kind; (c) * is of the first kind and N contains a symmetric root of unity. Without any assumption on the dimension of D or on the kind of *, the same conclusion holds in the cases: (d) N contains a symmetric root of unity whose minimal polynomial, in case k has positive characteristic, has even degree; and (e) N contains a symmetric element which is algebraic over k and whose minimal polynomial has degree 2.