Normal subgroups- Group theory

Mathematics presentation on the topic Normal Subgroups. B.sc 3rd semester topic.

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

4. 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 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 = Ng G = N G ( since, g = e and Ne = N ) N is a normal subgroup of G.  

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