The gap hypothesis for finite groups which have an Abelian quotient group not of order a power of 2.
Let \(G\) be a finite group and \(\pi(G)\) be the set of all prime divisors of \(|G|\). Put \[ \mathcal P(G)=\bigcup_{p\in\pi(G)}\mathcal P_p(G)\quad\text{and}\quad\mathcal L(G)=\bigcup_{p\in\pi(G)}\mathcal L_p(G), \] where \(\mathcal P_p(G)\) is the set of \(p\)-subgroups of \(G\) and \(\mathcal L_p(G)\) is the set of subgroups \(L\) of \(G\) which contain \(O^p(G)\), the smallest normal subgroup of \(G\) whose index is a power of \(p\), possibly 1, for a prime \(p\). Let \(V\) be a finite dimensional real \(G\)-representation space. \(V\) is called a `gap \(G\)-module' if \(V^L=0\) for all \(L\in\mathcal L(G)\) and \(\dim V^P>2\dim V^H\) for all \(P\in\mathcal P(G)\) and \(P<H\). A finite group \(G\) not of prime power order is called a `gap group' if there exists a gap \(G\)-module. In the paper under review the author gives a characterization of gap groups with \(O^{p_0}(G)\neq G\) for a unique odd prime \(p_0\). The two main results of the paper are as follows: Theorem 1.1. Suppose that \(\mathcal P(G)\cap\mathcal L(G)=\emptyset\), \(O^2(G)\neq G\) and \(O^{p_0}(G)\neq G\) for a unique odd prime \(p_0\). Then \(G\) is a gap group if and only if every subgroup \(K\) with \(O^2(G)\vartriangleleft K\leq G\) and \([K:O^2(G)]=2\) is a gap group. Theorem 1.2. Suppose that \(\mathcal P(G)\cap\mathcal L(G)=\emptyset\), \([G:O^2(G)]=2\) and \(O^{p_0}(G)\neq G\) for a unique odd prime \(p_0\). Then \(G\) is a gap group if and only if there is an element of \(G\) outside \(O^2(G)\) of order \(2^a\), \(a\geq 2\) or \[ \sum\frac{2}{|C_G(g)/O^2(C_G(g))|}<1, \] where the sum is taken over all representatives of conjugacy classes \(C_G(g)\) of elements \(g\) of \(G\) outside \(O^2(G)\) of order \(2\) such that \(O^2(C_G(g))\) is a \(p_0\)-group. Finally, the author gives some corollaries of the main results.
finite dimensional real representation spaces
scientific article; zbMATH DE number 6020909