For a finite noncyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H1,..., Hk with the property that every element of G is contained in Hig for some i∈{1,..., k} and g∈G. We prove that if G is a noncyclic permutation group of degree n, then γ(G)≤(n+2)/2. We then investigate the structure of the groups G with γ(G)=σ(G) (where σ(G) is the size of a minimal cover of G) and of those with γ(G)=2.
Covers and normal covers of finite groups
GARONZI, MARTINO;LUCCHINI, ANDREA
2015
Abstract
For a finite noncyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H1,..., Hk with the property that every element of G is contained in Hig for some i∈{1,..., k} and g∈G. We prove that if G is a noncyclic permutation group of degree n, then γ(G)≤(n+2)/2. We then investigate the structure of the groups G with γ(G)=σ(G) (where σ(G) is the size of a minimal cover of G) and of those with γ(G)=2.
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