Two point stabilisers of partition actions of linear groups. (English) Zbl 1156.20314
Summary: A partition of a \(kl\)-dimensional vector space \(V\) is a set of \(l\) subspaces each of dimension \(k\) such that their direct sum is the original space \(V\). We show that, unless \(l=2\), the action of a group \(\overline L\) such that \(\text{PSL}(V)\trianglelefteq\overline L\leq\text{PGL}(V)\) on the set of partitions of \(V\) into \(l\) subspaces of dimension \(k\) is base two: there exist two partitions \(\mathcal V\) and \(\mathcal W\) such that \(\overline L_{\mathcal V,\mathcal W}=1\).
We also show that, given any finite group \(G\), there exist \(k,l\) and partitions \(\mathcal{V,W}\) such that \(\overline L_{\mathcal V,\mathcal W}\cong G\).
These results complement work the author has done with partition actions of the symmetric groups.
We also show that, given any finite group \(G\), there exist \(k,l\) and partitions \(\mathcal{V,W}\) such that \(\overline L_{\mathcal V,\mathcal W}\cong G\).
These results complement work the author has done with partition actions of the symmetric groups.
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