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

Tian-Xiao He, Nikolai A. Krylov

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.

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

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 into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that cannot be embedded into a substitution group.

Paper Structure

This paper contains 3 sections, 7 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Truncated Riordan groups
  3. Impossibility of embedding $S_n\to {\cal R},n\geq 4$

Key Result

Lemma 1

The kernel of $P_0$ is isomorphic to the semidirect product where $\varphi: {\mathbb D}^*\to Aut({\mathbb D})$ is a homomorphism defined by $\varphi(a)(b) = \varphi_a(b) = ab$. The kernel of $P_n, n\geq 1$ is isomorphic to the direct product

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Claim 5
  • proof : Proof of the claim
  • ...and 6 more