Groups St Andrews 2005
Volume 1
Edited by C. M. Campbell
Edited by M. R. Quick
Edited by E. F. Robertson
Edited by G. C. Smith
Publisher: Cambridge University Press
Print Publication Year: 2007
Online Publication Date:May 2010
Online ISBN:9780511721212
Paperback ISBN:9780521694698
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511721212.009
Subjects: Algebra
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Abstract
Let p be a prime number, G be a finite p-group and K be a field of characteristic p. The Modular Isomorphism Problem (MIP) asks whether the group algebra KG determines the group G. Dealing with MIP, we investigated a question whether the nilpotency class of a finite p-group is determined by its modular group algebra over the field of p elements. We give a positive answer to this question provided one of the following conditions holds: (i) exp G = p; (ii) c1(G) = 2; (iii) G' is cyclic; (iv) G is a group of maximal class and contains an abelian subgroup of index p.
As a consequence, the positive solution of MIP for all p-groups containing a cyclic subgroup of index p2 was obtained.
Introduction
Though the Modular Isomorphism Problem is known for more than 50 years, up to now it remains open. It was solved only for some classes of p-groups, in particular:
pp. i-iv
pp. v-viii
pp. ix-x
Aspects of infinite permutation groups : Read PDF
pp. 1-35
Self-similarity and branching in group theory : Read PDF
pp. 36-95
On surface groups: motivating examples in combinatorial group theory : Read PDF
pp. 96-129
Nilpotent p-algebras and factorized p-groups : Read PDF
pp. 130-148
Classification of finite groups by the number of element centralizers : Read PDF
pp. 149-157
Algorithmic use of the Mal'cev correspondence : Read PDF
pp. 158-169
Minimal but inefficient presentations for semi-direct products of finite cyclic monoids : Read PDF
pp. 170-185
The modular isomorphism problem for finite p-groups with a cyclic subgroup of index p2 : Read PDF
pp. 186-193
On one-generated formations : Read PDF
pp. 194-198
New results on products of finite groups : Read PDF
pp. 199-202
Radical locally finite T-groups : Read PDF
pp. 203-206
Explicit tilting complexes for the Broué conjecture on 3-blocks : Read PDF
pp. 207-223
Conjugacy classes of p-regular elements in p-solvable groups : Read PDF
pp. 224-229
An algorithm for the unit group of the Burnside ring of a finite group : Read PDF
pp. 230-236
Integral group ring of the first Mathieu simple group : Read PDF
pp. 237-245
Embedding properties in direct products : Read PDF
pp. 246-255
Malcev presentations for subsemigroups of groups — a survey : Read PDF
pp. 256-268
Finite groups with extremal conditions on sizes of conjugacy classes and on degrees of irreducible characters : Read PDF
pp. 269-273
Conjugacy class structure in simple algebraic groups : Read PDF
pp. 274-287
On automorphisms of products of groups : Read PDF
pp. 288-305
Linear groups with infinite central dimension : Read PDF
pp. 306-312
G-automata, counter languages and the Chomsky hierarchy : Read PDF
pp. 313-318
An embedding theorem for groups universally equivalent to free nilpotent groups : Read PDF
pp. 319-326
Irreducible word problems in groups : Read PDF
pp. 327-340
Recent growth results : Read PDF
pp. 341-355