Normal subgroup generated by a subset

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Definition

Tabular definition

The normal subgroup generated by a subset, sometimes also called the normal closure of a subset, is defined in the following equivalent ways:

No. The normal subgroup generated by a subset is the ... The normal subgroup generated by a subset A of a group G is the ...
1 normal closure in the whole group of the subgroup generated by that subset normal closure \langle A \rangle^G where \langle A \rangle is the subgroup generated by A
2 smallest normal subgroup of the whole group that contains the subset smallest subgroup N \le G such that A \subseteq N and N is normal in G
3 subgroup generated by the set of all conjugate elements to elements of the subset subgroup \langle B \rangle where B = \bigcup_{g \in G} gAg^{-1}
4 the unique smallest possible kernel of a homomorphism from the whole group whose kernel contains the subset the smallest subgroup N containing A for which there is a homomorphism \varphi:G \to K such that the kernel of \varphi equals N. Any other subgroup arising as such a kernel must contain N.

The normal subgroup generated by a subset A of a group G is denoted \langle A^G \rangle, \langle A \rangle^G, or sometimes simply as A^G, though the final notation may also be used simply for the union of conjugates of A.

Facts

The normal subgroup generated by a subset depends on the ambient group, unlike the subgroup generated by a subset. In other words, if A is a subset of a group H which is a subgroup of a group G, the normal subgroup generated by A in H may differ from the normal subgroup generated by A in G.

Related notions