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⁻¹sg | 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 [math]\displaystyle{ \varnothing }[/math] 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.

References

• Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. pp. 73. ISBN 1-58488-372-3. https://archive.org/details/handbookofcomput0000holt/page/73. 
• 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.