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The modular isomorphism problem for the groups of order 512

Published online by Cambridge University Press:  05 July 2011

By
Bettina Eick  and
Alexander Konovalov
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Bettina Eick
Affiliation:
Technical University Braunschweig, Germany
Alexander Konovalov
Affiliation:
University of St Andrews
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

For a prime p, let G be a finite p-group and K a field of characteristic p. The Modular Isomorphism Problem (MIP) asks whether the modular group algebra KG determines the isomorphism type of G. We briefly survey the history of this problem and report on our computer-aided verification of the Modular Isomorphism Problem for the groups of order 512 and the field K with 2 elements.

Introduction

The Modular Isomorphism Problem has been known for more than 50 years. Despite various attempts to prove it or to find a counterexample to it, it is still open and remains one of the challenging problems in the theory of finite p-groups bordering on the theory of associative algebras.

Solutions for the modular isomorphism problem are available for various special types of p-groups. For example, the MIP holds for

  1. abelian p-groups (Deskins [14]; an alternative proof was given by Coleman [12]);

  2. p-groups G of class 2 with G′ elementary abelian (Sandling [34, Theorem 6.25]);

  3. metacyclic p-groups (Bagiński [1] for p > 3; completed by Sandling [36]);

  4. 2-groups of maximal class (Carlson [11]; alternative proof by Bagiński [3]);

  5. p-groups G of maximal class, p ≠ 2, where |G| ≤ pp+1 and G contains an abelian maximal subgroup (Caranti and Bagiński [2]);

  6. elementary abelian-by-cyclic groups (Bagiński [4]);

  7. p-groups with the center of index p2 (Drensky [16]); and

  8. p-groups having a cyclic subgroup of index p2 (Baginski and Konovalov [5]).

Information

Type
Chapter
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Groups St Andrews 2009 in Bath , pp. 375 - 383
Publisher: Cambridge University Press
Print publication year: 2011

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