Normal closure (group theory)

Smallest normal group containing a set From Wikipedia, the free encyclopedia

Normal closure (group theory)

In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing

Properties and description

Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing :[1]

The normal closure is the smallest normal subgroup of containing [1] in the sense that is a subset of every normal subgroup of that contains

The subgroup is the subgroup generated by the set of all conjugates of elements of in Therefore one can also write the subgroup as the set of all products of conjugates of elements of or their inverses:

Any normal subgroup is equal to its normal closure. The normal closure of the empty set is the trivial subgroup.[2]

A variety of other notations are used for the normal closure in the literature, including and

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 [3]

Group presentations

For a group given by a presentation with generators and defining relators the presentation notation means that is the quotient group where is a free group on [4]

References

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