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Abstract: A group is said to be covered by a collection of proper subgroups if each element of the group belongs to at least one subgroup in the collection: the collection of subgroups is called a covering of the group. If the set of subgroups is finite, we say the group has a finite covering. Many questions raised in the context of finite coverings of groups, such as conditions for a group to have a finite covering by abelian subgroups or determining the minimal number of subgroups needed to cover the group, make sense in other algebraic structures too.

In my talk I will report on my journeys through groups, loops, rings and semigroups, on what I discovered there about finite coverings together with several fellow travelers and on some discoveries which might still lie ahead.

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Eric Swartz

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