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.
"Gap modules for direct product groups." J. Math. Soc. Japan 53 (4) 975 - 990, October, 2001. https://doi.org/10.2969/jmsj/05340975