On small congruence covers (Q1125274)

Let \(G\) be a group of square order \(s ^2\). A congruence cover of \(G\) is a collection of \(r\) subgroups of \(G\) of order \(s\) the union of which is all of \(G\). The number \(e = r - s - 1\) is called the excess of the cover. Let \(p\) be the smallest prime divisor of \(s\) and assume that \(G\) admits a congruence cover of excess at most \(p-2\). Then \(G\) is elementary abelian and hence the cover is geometric, i. e. it can be viewed as a covering of a projective space by subspaces. For \(p\)-groups this bound is sharp; there are examples of non-elementary abelian \(p\)-groups admitting a congruence cover of excess \(p-1\).