Normal subgroups- Group theory
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Normal subgroups are subgroups where conjugation does not affect membership. A subgroup N of a group G is normal if gng-1 is in N for all g in G and n in N. A subgroup N is normal if and only if every left coset of N is also a right coset of N. If every left coset equals a right coset, then conjugation preserves membership in N, making N a normal subgroup.
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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, gng−1 ∈ N ∀ g∈ G, n ∈ N Equivalently, if gNg−1 = {gng−1 | n ∈ N}, then N is a normal subgroup of G if and only if gNg−1⊂ 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 gng−1= N ∀ g ∈ G (by theorem 1) (gng−1)g = Ng ∀ g ∈ G 0r gN (g−1g) = 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 g𝑁g−1 = Ngg−1 ∀ g ∈ G g𝑁g−1 = N ∀ g ∈ G ( since, gg−1 = e and Ne = N ) N is a normal subgroup of G.
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