On Generalized Heisenberg Groups: The Symmetric Case

In the literature, the famous Heisenberg group is the group of matrices of the form 1 x z 0 1 y 0 0 1 , where x , y , and z are real numbers. In the present article, we examine a generalized Heisenberg group, obtained from an R -module M endowed with an R -bilinear form β , where R is a ring with identity. We show that the structure of the generalized Heisenberg group and its generating space are intertwined. In particular, we prove that if β is symmetric, then the corresponding Heisenberg group possesses an involutive decomposition into subgroups, which eventually becomes the semidirect product of groups. This leads to a better understanding of the algebraic structure of the generalized Heisenberg group as well as its extensions by subgroups.