Date post: | 15-Apr-2017 |
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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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