Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 53, Number 4 (2001), 975-990.
Gap modules for direct product groups
Let be a finite group. A gap -module is a finite dimensional real -representation space satisfying the following two conditions:
(1) The following strong gap condition holds: for all such that is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a -surgery obstruction.
(2) has only one -fixed point 0 for all large subgroups , namely . A finite group not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product is a gap group for two finite groups and . According to , if and are gap groups, so is . In this paper, we prove that if is a gap group, so is . Using , this allows us to show that if a finite group has a quotient group which is a gap group, then itself is a gap group. Also, we prove the converse: if is not a gap group, then is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.
J. Math. Soc. Japan, Volume 53, Number 4 (2001), 975-990.
First available in Project Euclid: 29 May 2008
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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