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Math and science::Algebra::Aluffi

Normal subgroup

Normal subgroup

Let \( G \) be a group and \( N \) a subgroup of \( G \).

\( N \) is said to be normal iff:

\[ \forall n \in N, \forall g \in G, \; gng^{-1} \in N \]

Or equivalently:

\[ N \text{ is normal } \iff \forall g \in G, \; gN = Ng \]

\( gN \) is syntax for \( \{ a : a = gn \text{ for some } n \in N \} \), and \( gN = Ng \) does not imply commutativity.


Example

Abelian normal subgroup

Note how in the Abelian case, \( gng^{-1} = n \), which is a stronger result than \( gng^{-1} \in N \).

Non-abelian normal subgroup

Products have non-abelian normal subgroups.