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Siblings and twins in finite p-groups and a group identification for the groups of order $2^9$

Bettina Eick, Henrik Schanze

Abstract

We investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. We introduce the notations of siblings and twins for p-groups that are difficult to distinguish and we describe the siblings and twins among the groups of small prime-power order. We then use these ideas to device an effective group identification algorithm for the $10,494,213$ groups of order $2^9$.

Siblings and twins in finite p-groups and a group identification for the groups of order $2^9$

Abstract

We investigate which invariants of groups are powerful in distinguishing non-isomorphic p-groups. We introduce the notations of siblings and twins for p-groups that are difficult to distinguish and we describe the siblings and twins among the groups of small prime-power order. We then use these ideas to device an effective group identification algorithm for the groups of order .
Paper Structure (9 sections, 4 equations, 3 figures)

This paper contains 9 sections, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Siblings and twins
  3. Siblings and twins of order dividing $2^9$
  4. Group identification for the groups of order $2^9$
  5. Group identification for the groups of order $2^8$
  6. Group identification for the groups of order $2^9$
  7. Runtimes and bottlenecks
  8. Questions and open problems
  9. Appendix

Figures (3)

  • Figure 1: Average runtime of id-function.
  • Figure 2: Twins of order $2^9$, Part I
  • Figure 3: Twins of order $2^9$, Part II