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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:

[ \[ \forall n \in N, \forall g \in G, \; ? \] ]

Or equivalently:

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

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