Journal of the Mathematical Society of Japan

Gap modules for direct product groups

Toshio SUMI

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 29 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1212067582

Digital Object Identifier
doi:10.2969/jmsj/05340975

Mathematical Reviews number (MathSciNet)
MR1852892

Zentralblatt MATH identifier
1065.20004

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

Keywords
gap group gap module real representation direct product

Citation

SUMI, Toshio. Gap modules for direct product groups. J. Math. Soc. Japan 53 (2001), no. 4, 975--990. doi:10.2969/jmsj/05340975. https://projecteuclid.org/euclid.jmsj/1212067582


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