TY - JOUR

T1 - The Smith sets of finite groups with normal Sylow 2-subgroups and small nilquotients

AU - Koto, Akihiro

AU - Morimoto, Masaharu

AU - Qi, Yan

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - The Smith equivalence of real representations of a finite group has been studied by many mathematicians, e.g. J. Milnor, T. Petrie, S. Cappell-J. Shaneson, K. Pawałowski-R. Solomon. For a given finite group, let the primary Smith set of the group be the subset of real representation ring consisting of all differences of pairs of prime matched, Smith equivalent representations. The primary Smith set was rarely determined for a nonperfect group G besides the case where the primary Smith set is trivial. In this paper we determine the primary Smith set of an arbitrary Oliver group such that a Sylow 2-subgroup is normal and the nilquotient is isomorphic to the direct product of a finite number of cyclic groups of order 2 or 3. In particular, we answer to a problem posed by T. Sumi.

AB - The Smith equivalence of real representations of a finite group has been studied by many mathematicians, e.g. J. Milnor, T. Petrie, S. Cappell-J. Shaneson, K. Pawałowski-R. Solomon. For a given finite group, let the primary Smith set of the group be the subset of real representation ring consisting of all differences of pairs of prime matched, Smith equivalent representations. The primary Smith set was rarely determined for a nonperfect group G besides the case where the primary Smith set is trivial. In this paper we determine the primary Smith set of an arbitrary Oliver group such that a Sylow 2-subgroup is normal and the nilquotient is isomorphic to the direct product of a finite number of cyclic groups of order 2 or 3. In particular, we answer to a problem posed by T. Sumi.

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U2 - 10.1215/kjm/1250280981

DO - 10.1215/kjm/1250280981

M3 - Article

AN - SCOPUS:50949102014

VL - 48

SP - 219

EP - 227

JO - Journal of Mathematics of Kyoto University

JF - Journal of Mathematics of Kyoto University

SN - 0023-608X

IS - 1

ER -