The Groups <italic>G</italic>2<italic>n</italic> with Additional Structures.

In the paper [1], V. O. Manturov introduced the groups <italic>G</italic><italic>k</italic><italic>n</italic> depending on two natural parameters <italic>n</italic> > <italic>k</italic> and naturally related to topology and to the theory of dynamical systems. The group <italic>G</italic>2<italic>n</italic>, which is the simplest part of <italic>G</italic><italic>k</italic><italic>n</italic>, is isomorphic to the group of pure free braids on n strands. In the present paper, we study the groups <italic>G</italic>2<italic>n</italic> supplied with additional structures-parity and points; these groups are denoted by <italic>G</italic>2<italic>n</italic>,<italic>p</italic> and <italic>G</italic>2<italic>n</italic>,<italic>d</italic>. First,we define the groups <italic>G</italic>2<italic>n</italic>,<italic>p</italic> and <italic>G</italic>2<italic>n</italic>,<italic>d</italic>, then study the relationship between the groups <italic>G</italic>2<italic>n</italic>, <italic>G</italic>2<italic>n</italic>,<italic>p</italic>, and <italic>G</italic>2<italic>n</italic>,<italic>d</italic>. Finally, we give an example of a braid on <italic>n</italic> + 1 strands, which is not the trivial braid on <italic>n</italic> + 1 strands, by using a braid on n strands with parity. After that, the author discusses links in <italic>S</italic><italic>g</italic> × <italic>S</italic>1 that can determine diagrams with points; these points correspond to the factor S1 in the product <italic>S</italic><italic>g</italic> × <italic>S</italic>1. [ABSTRACT FROM AUTHOR]