nLab
normal closure

Given a group GG and a subset SGS\subset G, the normal closure of SS in GG is the smallest subgroup containing the set of all conjugates of elements in SS, i.e. the subgroup generated by the set of all elements of the form g 1sgg^{-1} s g, where gGg\in G and sSs\in S. The normal closure is clearly a normal subgroup of GG.

The normal closure, also called conjugate closure (see Wikipedia), should be distinguished from the normalizer of SS in GG. In combinatorial group theory a group GG is presented via specifying a set XX of generators and a set RR of relations. This means that G=F/NG=F/N where FF is the free group generated by XX and NN is the normal closure of RR in FF.

Revised on April 1, 2010 19:29:18 by Tim Porter (92.145.38.231)