In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.

Properties and description

Formally, if G is a group and S is a subset of G, the normal closure of S is the intersection of all normal subgroups of G containing S: The normal closure is the smallest normal subgroup of G containing S, in the sense that is a subset of every normal subgroup of G that contains S. The subgroup is generated by the set of all conjugates of elements of S in G. Therefore one can also write Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set \varnothing is the trivial subgroup. A variety of other notations are used for the normal closure in the literature, including and Dual to the concept of normal closure is that of or, defined as the join of all normal subgroups contained in S.

Group presentations

For a group G given by a presentation with generators S and defining relators R, the presentation notation means that G is the quotient group where F(S) is a free group on S.