Normal Closure (other)
The term ''normal closure'' is used in two senses in mathematics: * In group theory, the normal closure (group theory), normal closure of a subset of a group is the smallest normal subgroup that contains the subset. * In field theory, the Normal closure (field theory), normal closure of an algebraic extension ''F''/''K'' is an extension field ''L'' of ''F'' such that ''L''/''K'' is normal and ''L'' is minimal with this property. {{disambiguation ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Normal Closure (group Theory)
In group theory, the normal closure of a subset S of a Group (mathematics), 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 \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 Generating set of a group, generated by the set S^G=\ = \ of all Conjugacy class, 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. A variety of other notations are used for the normal closure in the literature, including \langle S^G\rangle, \langle S ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]