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A **normal subgroup** is a subgroup that is invariant under conjugation by any element of the original group: H is normal if and only if gHg^{-1} = H for any g âˆˆ G. Equivalently, a subgroup H of G is normal if and only if gH = Hg for any g âˆˆ G.

Normal subgroups are useful in constructing quotient groups, and in analyzing homomorphisms.

**Quotient Groups**

A **quotient group** is defined as G/N for some normal subgroup N of G, which is the set of cosets of N w.r.t. G, equipped with the operation Âº satisfying for all g, h âˆˆ G.

This definition is the reason that N must be normal to define a quotient group; it holds because the chain of equalities

holds, where utilizes the fact that Nh = hN for any h (true iff N is normal, by definition).

For example, consider the subgroup (which is an additive group). The left cosets are

so This can be more cleanly written as

which is isomorphic to {0,1} or the cyclic group C_{2}. Additional examples:

- The quotient group Z/2Z, where 2Z--t the group of even integers--is a normal subgroup of , is isomorphic to as well.
- The quotient group R/Z, where Z--the group of integers--is a normal subgroup of the reals , is isomorphic to the
**circle group**defined by the complex numbers with magnitude 1.

H**omomorphisms and Normal Subgroups**

Recall that a homomorphism from G to H is a function Ï† such that

for all g_{1}, g_{2} âˆˆ G.

The **kernel** of a homomorphism is the set of elements of G that are sent to the identity in H, and the kernel of any homomorphism is necessarily a normal subgroup of G.

In fact, more is true: the **image** of G under this homomorphism (the set of elements G is sent to under Ï†) is isomorphic to the quotient group G/ker(Ï†), by the **first isomorphism theorem**. This provides a bijection between normal subgroups of G and the set of images of G under homomorphisms.

Thus normal subgroups can be classified in another manner:

A subgroup N of G is normal if and only if there exists a homomorphism on G whose kernel is N. |

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