Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02
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Equivalent characterisations of a normal subgroup by conjugates and left and right cosets

Statement

Let N≤G. The following conditions are equivalent:

  1. N⊴G (Normal subgroup: invariance under conjugation);
  2. gNg−1⊆N for every g∈G;
  3. gN=Ng for every g∈G, where these are the left and right cosets of N represented by g.

Facts & Assumptions

Given: A group G and a subgroup N≤G.

[F1]

The subgroup N is normal when gNg−1=N for every g∈G (Normal subgroup: invariance under conjugation).

[L1]

In a group, (g−1)−1=g and (gh)−1=h−1g−1 (In a group e−1=e, (g−1)−1=g and (gh)−1=h−1g−1, the order of the last product being essential).

Proof

technique · direct
1.1

Condition 1 implies condition 2 because equality implies containment.

F1
1.2

Suppose condition 2 holds. Applying it to g−1 gives g−1Ng⊆N; conjugating this containment by g and using (g−1)−1=g gives N⊆gNg−1, while condition 2 gives the reverse containment. Hence gNg−1=N for every g, so condition 1 holds.

givenL1algebra
1.3

Suppose condition 1 holds. If x∈gN, then x=gn=(gng−1)g for some n∈N, and gng−1∈N by [F1], so x∈Ng. Replacing g by g−1 gives the reverse inclusion, hence gN=Ng.

givenF1L1algebra
1.4

Suppose condition 3 holds. For n∈N, the element gn lies in gN=Ng, so gn=n′g for some n′∈N; therefore gng−1=n′∈N. Thus gNg−1⊆N and condition 2 holds.

givenalgebra
2.1

Steps 1.1 through 1.4 prove that conditions 1, 2, and 3 are equivalent.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

Dependency tree · two levels

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Sources