Twisted virtual braid group

In this paper, we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group [Formula: see text]. In particular, the twisted virtual pure braid group [Formula: see text] is the kernel of an epimorphism of [Formula: see text] onto the symmetric group [Formula: see text]. We find the set of generators and defining relations for [Formula: see text] and show that [Formula: see text]. Further, we prove that [Formula: see text] is a semi-direct product of some subgroup and abelian group [Formula: see text]. As a corollary, we get that the virtual pure braid group [Formula: see text] is a subgroup of [Formula: see text]. Also, we construct some other epimorphism of [Formula: see text] onto [Formula: see text]. Its kernel [Formula: see text] is an analogous of [Formula: see text]. We find its set of generators and define relations and construct their decomposition in a semi-direct product.