Normal subgroup

Correspondence between normal subgroups and congruence relations

A Normal subgroup 𝑁 ◃𝐺 uniquely defines a congruence relation ≡ on 𝐺 and vice versa, group such that [𝑒]≡ =𝑁, and for any 𝑔 ∈𝐺 the congruence classes correspond to cosets of 𝑁:

[𝑔]≡=𝑔𝑁=𝑁𝑔

Proof

First, we will prove that [𝑒]≡ is a subgroup. Clearly 𝑒 ∈[𝑒]≡. Let 𝑎,𝑏 ∈[𝑒]≡, i.e. 𝑎 ≡𝑏 ≡𝑒. Then 𝑎𝑏−1 ≡𝑏𝑏−1 =𝑒 and therefore 𝑎𝑏−1 ∈[𝑒]≡. Therefore [𝑒]≡ is a subgroup by One step subgroup test.

Next, we will show that 𝑁 =[𝑒]≡ is a Normal subgroup. Let 𝑔 ∈𝐺 and 𝑛 ∈𝑁. Then 𝑔𝑛𝑔−1 ≡𝑔𝑒𝑔−1 =𝑒 and thus 𝑔𝑛𝑔−1 ∈𝑁. Hence 𝑔𝑁𝑔−1 =𝑁 for any 𝑔 ∈𝐺, Therefore 𝑁 is normal.

Finally we show the equivalence between (left) cosets of 𝑁 and congruence classes. For any 𝑔,ℎ ∈𝐺

ℎ∈[𝑔]≡⟺ℎ≡𝑔⟺𝑔−1ℎ≡𝑒⟺𝑔−1ℎ∈𝑁⟺ℎ∈𝑔𝑁

as required.

As a result of this theorem, normal subgroups may be used to form a Quotient group (following the usual notion of Algebraic quotient) where each coset is taken as a group element.

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