The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group

TL;DR

This paper demonstrates that large symmetric and non-abelian simple groups cannot be embedded into Riordan groups, highlights the solvability of truncated Riordan groups, and provides specific embeddings and obstructions for certain groups.

Contribution

It establishes non-embeddability results for symmetric and simple groups into Riordan groups and explores the structure of truncated Riordan groups and specific group embeddings.

Findings

Symmetric groups of degree > 3 are not embeddable in Riordan groups.

Finite non-abelian simple groups cannot be embedded into Riordan groups.

All truncated Riordan groups are solvable.

Abstract

We prove that the symmetric group of degree greater than three cannot be embedded into the Riordan group with coefficients in any commutative ring. We also prove the impossibility to embed finite non-abelian simple groups. As a closely related topic, we show why all truncated Riordan groups are solvable, in stark contrast to the unsolvability of the infinite-sized Riordan groups. Finally, we give an explicit embedding of the alternating group $A_4$ into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that $A_4$ cannot be embedded into a substitution group.