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Concept# Normal subgroup

Summary

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is
Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of are precisely the kernels of group homomorphisms with domain which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
A subgroup of a group is called a normal subgroup of if it is invariant under conjugation; that is, the conjugation of an element of by an element of is always in The usual notation for this relation is
For any subgroup of the following conditions are equivalent to being a normal subgroup of Therefore, any one of them may be taken as the definition.
The image of conjugation of by any element of is a subset of i.e., for all .
The image of conjugation of by any element of is equal to i.e., for all .
For all the left and right cosets and are equal.
The sets of left and right cosets of in coincide.
Multiplication in preserves the equivalence relation "is in the same left coset as". That is, for every satisfying and , we have
There exists a group on the set of left cosets of where multiplication of any two left cosets and yields the left coset . (This group is called the quotient group of modulo , denoted .)
is a union of conjugacy classes of
is preserved by the inner automorphisms of
There is some group homomorphism whose kernel is
There exists a group homomorphism whose fibers form a group where the identity element is and multiplication of any two fibers and yields the fiber . (This group is the same group mentioned above.)
There is some congruence relation on for which the equivalence class of the identity element is .

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