A study of representations and finite group actions realizing a given fixed point set

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Let G be a finite group not of prime power order. For a prime p, we denote by O^p(G), called the Dress group of type p, the smallest normal subgroup of G with p-power index. The group G is said to be a gap group if there exists a G-module V such that dim V^<O^2(G)>=0 for any prime p and dim V^P>2dim V^H for any pair (P,H) of subgroups of G with some condition. Note that a Dress subgroup of a gap group G is not of prime power order.I show that G is a gap group if and only if all subgroups L of G, possessing cyclic quotients L/O^2(G), are gap groups. Now assume that G/O^2(G) is cyclic. As viewing the series of normal groupsG=G_0〓G_1〓G_2〓【triple bond】〓G_k=O^2(G),[G_j,G_<j-1>]=2,I obtained that G is a gap group if and only if each G_j(0<j<k) is a gap group. I define a subset E_j of 2-elements h of G_j＼G_<j-1> by using a form of the centralizer C_G(h). We can easily decide whether the set E_j is empty or not, for example, letting j>1,E_j is not empty if there exists an element of G_j＼G_<j-1> not of prime power order. It is a little bit complication to decide whether E_1 is empty. Then I showed that all E_j are nonempty if and only if G is a gap group. Furthermore, I obtained that if G×【triple bond】×G is a gap group,then so is G×G.