Journal of the Mathematical Society of Japan

Gap modules for direct product groups

Toshio SUMI

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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.

Article information

J. Math. Soc. Japan, Volume 53, Number 4 (2001), 975-990.

First available in Project Euclid: 29 May 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S17: Finite transformation groups 20C15: Ordinary representations and characters

gap group gap module real representation direct product


SUMI, Toshio. Gap modules for direct product groups. J. Math. Soc. Japan 53 (2001), no. 4, 975--990. doi:10.2969/jmsj/05340975.

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