conjugate closure
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In
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the normal closure of a subset S of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 \operatorname_G(S) of S is the intersection of all normal subgroups of G containing S: \operatorname_G(S) = \bigcap_ N. The normal closure \operatorname_G(S) is the smallest normal subgroup of G containing S, in the sense that \operatorname_G(S) is a subset of every normal subgroup of G that contains S. The subgroup \operatorname_G(S) is generated by the set S^G=\ = \ of all conjugates of elements of S in G. Therefore one can also write \operatorname_G(S) = \. Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set \varnothing is the
trivial subgroup In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
. A variety of other notations are used for the normal closure in the literature, including \langle S^G\rangle, \langle S\rangle^G, \langle \langle S\rangle\rangle_G, and \langle\langle S\rangle\rangle^G. 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 A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
G=\langle S \mid R\rangle with generators S and defining relators R, the presentation notation means that G is the quotient group G = F(S) / \operatorname_(R), where F(S) is a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
on S.


References

Group theory Closure operators {{Abstract-algebra-stub