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On the isomorphism problem for certain $p$-groups

Alexander Montoya Ocampo, Fernando Szechtman

Abstract

We consider 9 infinite families of finite $p$-groups, for $p$ a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified.

On the isomorphism problem for certain $p$-groups

Abstract

We consider 9 infinite families of finite -groups, for a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified.
Paper Structure (11 sections, 23 theorems, 151 equations)

This paper contains 11 sections, 23 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Sufficiency
  3. Searching for isomorphisms
  4. Necessity for $J_1(\alpha)$
  5. Necessity for $J_2(\alpha)$ Part I
  6. Necessity for $H_2(\alpha)$
  7. Necessity for $J_2(\alpha)$ Part II
  8. Common structural features of $J_1(\alpha)$ and $J_1(\alpha')$
  9. Common structural features of $J_2(\alpha)$ and $J_2(\alpha')$
  10. Proof of Theorems A through E
  11. Appendix

Key Result

Theorem 2.1

(a) In Case 1, assume $\ell'\equiv \ell\mod p^{m}$, i.e., $\alpha'\equiv\alpha\mod p^{2m}$. We then have $J_1(\alpha')\cong J_1(\alpha)$. (b) In Case 2, suppose that $\ell'\equiv \ell\mod 2^{m}$, that is, $\alpha'\equiv\alpha\mod 2^{2m}$. Then $J_2(\alpha')\cong J_2(\alpha)$. (c) In Case 3, $J_3(\al

Theorems & Definitions (46)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 36 more