NORMAL SUBGROUPS

Presentation byDurwas

Maharwade

Definition:A subgroup N of a group G is said to be a normal subgroup of G if, N G, n N

Equivalently, if = {n N}, then N is a normal subgroup of G if and only if ⊂ N g G.

Theorem 2The subgroup N of a group G is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G.Proof :Let N be a normal subgroup of G.

Then = N G (by theorem 1) ()g = Ng G

0r gN (g) = Ng G

gN = Ng G

i.e., every left coset gN is the right coset Ng.

Conversely, assume that every left coset of a subgroup N of G is the right coset of N in G.

Thus, for G , a left coset gN must be a right coset.

Nx for some xG.

Now, e N ge = g gN.

g Nx ( since gN = Nx )

Also, g = eg Ng, a right coset of N in G.

Thus, two right cosets Nx and Ng have common element g.

Nx = Ng ( since two right cosets are either identical or disjoint.)

Ng is the unique right coset which is equal to the left coset gN.

gN = Ng G

= Ng G

= N G ( since, g = e and Ne = N )

N is a normal subgroup of G.

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