On the size of coset unions

Abstract

AbstractLet $$g_1H_1,\ldots ,g_nH_n$$ g 1 H 1 , … , g n H n be cosets of subgroups $$H_1,\ldots ,H_n$$ H 1 , … , H n of a finite group G such that $$g_1H_1\cup \ldots \cup g_nH_n\ne G$$ g 1 H 1 ∪ … ∪ g n H n ≠ G . We prove that $$|g_1H_1\cup \ldots \cup g_nH_n|\le \gamma _n|G|$$ | g 1 H 1 ∪ … ∪ g n H n | ≤ γ n | G | where $$\gamma _n<1$$ γ n < 1 is a constant depending only on n. In special cases, we show that $$\gamma _n=(2^n-1)/2^n$$ γ n = ( 2 n - 1 ) / 2 n is the best possible constant with this property and we conjecture that this is generally true.