Normal subgroups- Group theory
- 1. NORMAL SUBGROUPS Presentation by Durwas Maharwade
- 2. Definition: A subgroup N of a group G is said to be a normal subgroup of G if, gng1 N g G, n N Equivalently, if gNg1 = {gng1 | n N}, then N is a normal subgroup of G if and only if gNg1 N g G.
- 3. Theorem 2 The 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 gng1= N g G (by theorem 1) (gng1)g = Ng g G 0r gN (g1g) = Ng g G gN = Ng g G i.e., every left coset gN is the right coset Ng.
- 4. Conversely, assume that every left coset of a subgroup N of G is the right coset of N in G. Thus, for g G , a left coset gN must be a right coset. gN = Nx for some x G. Now, e N ge = g gN. g Nx ( since gN = Nx ) Also, g = eg Ng, a right coset of N in G.
- 5. 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 G gg1 = Ngg1 g G gg1 = N g G ( since, gg1 = e and Ne = N ) N is a normal subgroup of G.
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