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Let $ G $ be a finite group not of prime power order. A gap $ G $-module $ V $ is a finite-dimensional real $ G $-representation space satisfying the following two conditions. The first is the conditi...on dim $ V^p > 2 $ dim $ V^H $ for all $ P < H leqq G $ such that $ P $ is of prime power order and the other is the condition that $ V $has only one $ H $-fixed point $ 0 $ for all large subgroups $ H $ $ colon $ precisely to say, $ H in L(G) $. If there exists a gap $ G $-module, then $ G $ is called a gap group. We study $ G $-modules induced from $ C $-modules for subgroups $ C $ of $ G $ and obtain a sufficient condition for $ G $ to become a gap group. Consequently, we show that non-solvable general linear groups and the automorphism groups of sporadic groups are all gap groups.続きを見る
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