Normal subgroup

In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written:

Related Topics:
Mathematics - Group - Subgroup - Conjugation

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:N riangleleft G.

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There are a number of conditions which are equivalent to requiring that a subgroup N be normal in G. Any one of them may be taken as the definition:

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• For all g in G, g⁻¹Ng ⊆ N.
• For all g in G, g⁻¹Ng = N.
• The sets of left and right cosets of N in G coincide.
• For each g in G, gN = Ng.
• N is a union of conjugacy classes of G.

Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4). For this reason, conditions (1) and (3) are often used to prove that N is normal in G, while conditions (2) and (4) are used to prove consequences of the normality of N in G.

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All subgroups N of an abelian group G are normal, because g−1(Ng) = g−1(gN) = (g−1g)N = N. A group that

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is not Abelian but for which every subgroup is normal is termed a Hamiltonian group.

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The normal subgroups of any group G form a lattice under inclusion. The minimum and maximum elements are {e} and G, the greatest lower bound of two normal subgroups is their intersection and their least upper bound is a product group.

Related Topics:
Lattice - Product group

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Galois was the first to realize the importance of the existence of normal subgroups.

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Normal subgroup: Example ►