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Abstract Algebra

Exercise 3.1.39 ( $D$ is not normal in $A \times A$)

Suppose A is the non-abelian group S 3 and D the diagonal subgroup { ( a , a ) ∣ a ∈ A } . Prove that D is not normal in A × A .

Answers

Proof. For all σ 1 , σ 2 , τ ∈ S 3 ,

( σ 1 , σ 2 ) ( τ , τ ) ( σ 1 , σ 2 ) − 1 = ( σ 1 τ σ 1 − 1 , σ 2 τ σ 2 − 1 ) .

To build a counterexample, we choose

τ = ( 1 2 ) σ 1 = ( ) = id [ [ 1 , n ] ] σ 2 = ( 1 2 3 ) .

Then σ 1 τ σ 1 − 1 = τ , and

σ 2 τ σ 2 − 1 = ( 1 2 3 ) ( 1 2 ) ( 1 2 3 ) − 1 = ( 2 3 ) ≠ ( 1 2 ) .

Therefore, for this choice of σ 1 , σ 2 , τ ∈ S 3 , even though ( τ , τ ) ∈ D ,

( σ 1 , σ 2 ) ( τ , τ ) ( σ 1 , σ 2 ) − 1 = ( ( 1 2 ) , ( 2 3 ) ) ∉ D .

This shows that D is not a normal subgroup of S 3 × S 3 . □

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2025-12-06 11:56
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