# Conjugate closure

In group theory, the **conjugate closure** of a subset *S* of a group *G* is the subgroup of *G* generated by *S*^{G}, i.e. the closure of *S*^{G} under the group operation, where *S*^{G} is the set of the conjugates of the elements of *S*:

*S*^{G}= {*g*^{−1}*sg*|*g*∈*G*and*s*∈*S*}

The conjugate closure of *S* is denoted <*S*^{G}> or <*S*>^{G}.

The conjugate closure of any non-empty subset *S* of a group *G* is always a normal subgroup of *G*; in fact, it is the smallest (by inclusion) normal subgroup of *G* which contains the non-empty set *S*. For this reason, the conjugate closure coincides with the **normal closure** of *S* or the **normal subgroup generated by** *S* for non-empty *S*, where the normal closure is defined as the intersection of all normal subgroups of *G* which contain *S*. Any normal subgroup is equal to its normal closure.

The conjugate closure of a singleton subset {*a*} of a group *G* is a normal subgroup generated by *a* and all elements of *G* which are conjugate to *a*. Therefore, any simple group is the conjugate closure of any non-identity group element. Note that the conjugate closure of the empty set is empty, while its normal closure is the trivial group.

Contrast the normal closure of *S* with the *normalizer* of *S*, which is (for *S* a group) the largest subgroup of *G* in which *S* *itself* is normal. (This need not be normal in the larger group *G*, just as <*S*> need not be normal in its conjugate/normal closure.)

Dual to the concept of normal closure is that of *normal interior* or *normal core*, defined as the join of all normal subgroups contained in *S*.[1]

## References

- Robinson p.16

- Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005).
*Handbook of Computational Group Theory*. CRC Press. p. 73. ISBN 1-58488-372-3. - Robinson, Derek J. S. (1996).
*A Course in the Theory of Groups*. Graduate Texts in Mathematics.**80**(2nd ed.). Springer-Verlag. ISBN 0-387-94461-3. Zbl 0836.20001.