# Representation Theory, Group Rings, and Coding Theory

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*M. Isaacs; A. Lichtman; D. Passman; S. Sehgal; N. J. A. Sloane; H. Zassenhaus*

This volume is dedicated to the memory of the Soviet mathematician S. D. Berman (1922-1987). Berman's work—for the most part in representation theory, group rings, and coding theory—is discussed here in a number of review articles. Among the topics covered are Berman's achievements in coding theory, including his pioneering work on abelian codes and his results on the theory of threshold functions. Also discussed are his contributions to the representation theory of groups over fields, his work on integral representations of groups, his accomplishments in infinite abelian group rings, and his fundamental results on units in integral group rings. In addition, there are 22 research articles written by an international group of researchers in areas of Berman's major interest.

# Table of Contents

## Representation Theory, Group Rings, and Coding Theory

- Contents xi12 free
- Preface xv16 free
- S.D. Berman (1922–1987): A Biographical Note xvii18 free
- Publications of S. D. Berman xxv26 free
- Ph.D. Students of S.D. Berman xxxi32 free
- The S. D. Berman work on coding theory and on theory of threshold functions 134 free
- Berman's contributions to group representation over fields 1750
- S. D. Berman's contribution to the theory of integral representations of finite groups 2760
- The work of S. D. Berman on infinite abelian group rings 3972
- Berman's work on units in group rings 4780
- On p-subgroups of finite symmetric and alternating groups 67100
- Parameters of abelian codes in the group algebra KG of G = (a) x (b), aP = bP = 1, p prime, over a finite field K with primitive pth root of unity and related MDS-codes 77110
- Locally cyclic modules over a valuation ring 85118
- On the modules over a real crossed group algebra 105138
- The minimal number of generators of wreath products of nilpotent groups 115148
- Torsion free space groups and permutation lattices for finite groups 123156
- Clifford theory and induction from subgroups 133166
- The isomorphism problem for modular group algebras of groups with large centres 145178
- Zero divisors in group rings: Something old, something new 155188
- The augmentation quotients of finite abelian p-groups 167200
- Free groups in normal subgroups of unit groups and arithmetic groups 173206
- On the Bass-Milnor index of abelian p-groups 179212
- A remark on algebraic geometric codes 197230
- On decomposition of a binomial in a skew polynomial ring 201234
- The associated graded algebra of a group algebra 209242
- On nilpotent and soluble subgroups of linear groups over fields of fractions of enveloping algebras and of group rings, I 247280
- The structure of Go for certain polycyclic group algebras and related algebras 283316
- The direct factor problem for modular abelian group algebras 303336
- On the isomorphism of the semi-simple group algebras of the primary abelian groups 309342
- On the canonical irreducible representations of countable locally finite groups and infinite Frobenius group 321354
- Generators of subgroups of U(ZG) 331364
- Connectivity of the finite group and generalized characters 349382