On the closure of an elementary Abelian 2-subgroup of a finite group. (Q1860910)

Let \(k\) be a field of two elements, let \(H\) be a finite group, let \(S\in\text{Syl}_2(H)\), let \(V\) be a \(kH\)-module (with \(| V|\) finite) and let \(t\) be an integer. Set \[ \begin{multlined}\mu_t(S)=\{ U\leq V\mid U\text{ is }S\text{-invariant and }U=\langle u\in U\mid\dim_k([u,A])\leq t\\ \text{for any elementary Abelian subgroup }A\text{ of }S\text{ with }[U,A,A]=0\rangle\}.\end{multlined} \] Suppose that \(U\in\mu_t(S)\) and let \(U^H=\langle u^h\mid u\in U,\;h\in H\rangle\). Here restrictive hypotheses A and B are required. The two main results in this situation, of this note, can be summarized as follows: (1) Assuming Hypothesis A, then \(U^H\in\mu_{t+1}(S)\) (Theorem 1); and (2) Assuming Hypothesis B, then an integer \(m\) is defined such that \(U^H\in\mu_{t+m}(S)\) or \(U^H\in\mu_{t+m+1}(S)\) and several other conditions hold (Theorem 2).