On a set of normal subgroups

Abstract

The commutator [ a, b ] of two elements a and b in a group G satisfies the identity ab = ba [ a, b ]. The subgroups we study are contained in the commutator subgroup G ′, which is the subgroup generated by all the commutators. The group G is covered by a well-known set of normal subgroups, namely the normal closures { g } G of the cyclic subgroups { g } in G . In a similar way one may associate a subgroup K ( g ) with each element g , by defining K ( g ) to be the subgroup generated by the commutators [ g, x ] as x takes all values in G . These subgroups generate G ′ (but do not cover G ′ in general), and are normal in G in consequence of the identical relation (A) [ g, x ] Y = [ g, y ] −1 [ g, xy ] holding for all g, x and y in G . (By a b we mean b −1 ab .) It is easy to see that { g } G = { g , K ( g )}.

Locations

• Proceedings of the Glasgow Mathematical Association - View - PDF