The group generated by Riordan involutions

TL;DR

This paper characterizes the subgroup generated by Riordan involutions, proving that any element can be expressed as a product of at most four involutions and describing its algebraic structure.

Contribution

It provides a new structural description of the subgroup generated by Riordan involutions, including bounds on element decompositions and a semidirect product characterization.

Findings

Any element in the subgroup is a product of at most four Riordan involutions

The subgroup is a semidirect product of a specific commutator subgroup and the Klein four-group

Structural insights into the algebraic composition of Riordan involutions

Abstract

We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.