Normal subgroup: Difference between revisions

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* [http://mathworld.wolfram.com/NormalSubgroup.html Normal subgroup] on Mathworld
* [http://mathworld.wolfram.com/NormalSubgroup.html Normal subgroup] on Mathworld
* [http://planetmath.org/encyclopedia/NormalSubgroup.html Normal subgroup] on Planetmath
* [http://planetmath.org/encyclopedia/NormalSubgroup.html Normal subgroup] on Planetmath
* [http://en.wikipedia.org/wiki/Normal_subgroup Normal subgroup] on Wikipedia

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Definition

A subgroup H of a group G is termed normal if the following equivalent conditions are satisfied:

  1. Given any and , we have
  2. H occurs as the kernel of a homomorphism from G. In other words, there is a homomorphism such that the inverse image of the identity element of K is H.
  3. Every inner automorphism of G sends H to within itself
  4. Every inner automorphism of G restricts to an automorphism of H

Some elementary examples and nonexamples

All subgroups in Abelian groups

In an Abelian group, every subgroup is normal. This is because if is an Abelian group, and , then .

More generally, any subgroup inside the center of a group is normal.

It is not, however, true that if every subgroup of a group is normal, then the group must be Abelian. A counterexample is the quaternion group.

All characteristic subgroups

A characteristic subgroup of a group is a subgroup which is invariant under all automorphisms of the whole group. Characteristic subgroups are normal, because normality requires invariance only under inner automorphisms, which are a particular kind of automorphism.

In particular, subgroups like the center, the commutator subgroup, the Frattini subgroup are examples of characteristic, and hence normal, subgroups.

A smallest non-example

The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle in the symmetric group of permutations on symbols .

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