Normal subgroup Totally Explained

Everything about Normal Subgroup totally explained

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group. Évariste Galois was the first to realize the importance of the existence of normal subgroups.

Definitions

A subgroup N of a group G is called a normal subgroup if it's invariant under conjugation; that is, for each element n in N and each g in G, the element gng⁻¹ is still in N. We write ť

N 	riangleleft G,,Leftrightarrow,forall,nin in

H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f) (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups G/N of G and the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we've shown that the normal subgroups are precisely the kernels of homomorphisms with domain G.

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