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October, 2001 Gap modules for direct product groups
Toshio SUMI
J. Math. Soc. Japan 53(4): 975-990 (October, 2001). DOI: 10.2969/jmsj/05340975


Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions:

(1) The following strong gap condition holds: dimVP>2dimVH for all P<HG such that P is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a G-surgery obstruction.

(2)V has only one H-fixed point 0 for all large subgroups H, namely HL(G). A finite group G not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product K×L is a gap group for two finite groups K and L. According to [5], if K and K×C2 are gap groups, so is K×L. In this paper, we prove that if K is a gap group, so is K×C2. Using [5], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K×D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.


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Toshio SUMI. "Gap modules for direct product groups." J. Math. Soc. Japan 53 (4) 975 - 990, October, 2001.


Published: October, 2001
First available in Project Euclid: 29 May 2008

zbMATH: 1065.20004
MathSciNet: MR1852892
Digital Object Identifier: 10.2969/jmsj/05340975

Primary: 20C15 , 57S17

Keywords: direct product , gap group , gap module , real representation

Rights: Copyright © 2001 Mathematical Society of Japan


Vol.53 • No. 4 • October, 2001
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