This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Since the equivalence relation induced by being in the same coset of the normal subgroup is a congruence, the coset space can actually be equipped with a canonical group structure, and the quotient map then becomes a surjective group homomorphism.
The coset space with this group structure is also termed the quotient group.
Definition with symbols
The equivalence relation induced by the cosets of a normal subgroup is a congruence, viz.:
Thus, we can define a group structure on the coset space by setting (this is well-defined and independent of the choice of coset representatives precisely because of the above fact).
Note that the quotient map is a surjective homomorphism whose kernel is the given normal subgroup.
The group is also termed the quotient group of via this quotient map.
- Normal subgroup equals kernel of homomorphism: The kernel of any homomorphism is a normal subgroup. Conversely, any normal subgroup is a kernel of a homomorphism, namely the quotient map for that normal subgroup.
- First isomorphism theorem: This states that for any homomorphism of groups with kernel , the map from , restricted to the image, is equivalent to the quotient map
- Second isomorphism theorem
- Third isomorphism theorem
- Fourth isomorphism theorem