Definition: Normal Subgroups

A subgroup \(N\) of a group \(G\) is called normal and denoted as \(N\unlhd G\), if and only if:

(1) its left and right cosets with respect to any element \(a\in G\) coincide, i.e. \[aN=Na,~\forall a\in G.\]

Equivalent definition:

(2) for all \(a\in G\): \[aNa^{-1}=N\]

where the set \(aNa^{-1}\) is defined as \(\{g\in G~|~g=a\ast h\ast a^{-1},~h\in N\}.\)

Lemmas: 1 2
Proofs: 3 4 5 6 7

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  1. Ayres jr. Frank: "Theory and Problems of Modern Algebra", McGraw-Hill Book Company Europe, 1978