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Coverings of groups by subgroups

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If you have a question about this talk, please contact David Craven.

Let G be a finite group. A covering of G is a family of proper subgroups of G whose union is G. Such a family exists if and only if G is non-cyclic. If G is non-cyclic define s(G) to be the smallest cardinality of a covering of G, and set s(G) equal to infinity if G is cyclic. The function s was introduced by J.H.E. Cohn and has been studied by several authors. Let N be a normal subgroup of G. It is clear that any covering of G/N can be lifted to a covering of G of the same cardinality, and this implies that s(G) is bounded from above by s(G/N). If there exists N such that s(G) = s(G/N) then we are reduced to study the quotient G/N, and for this reason we are very much interested in the groups G with the property that s(G) < s(G/N) for any non-trivial normal subgroup N of G (these groups are traditionally called ‘sigma-primitive’).

Lucchini and Detomi conjectured that such groups either are abelian or admit exactly one minimal normal subgroup. Proving this would improve very much our understanding of the function s. In this talk we present this conjecture and the progress made so far about it.

This talk is part of the Algebra Seminar series.

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