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Abstract Algebra

Exercise 2.4.6 (Subgroup $\langle (1\ 2), (1\ 2) (3\ 4) \rangle \leq S_4$)

Prove that the subgroup of S 4 generated by ( 1 2 ) and ( 1 2 ) ( 3 4 ) is a non cyclic group of order 4 .

Answers

Proof. Let H = ⟨ ( 1 2 ) , ( 1 2 ) ( 3 4 ) ⟩ . Then

( 3 4 ) = ( 1 2 ) [ ( 1 2 ) ( 3 4 ) ] ∈ H ,

thus

K = { e , ( 1 2 ) , ( 3 4 ) , ( 1 2 ) ( 3 4 ) } ⊆ H .

Since the cycles ( 1 2 ) and ( 3 4 ) are disjoint, they commute, thus ( 3 4 ) [ ( 1 2 ) ( 3 4 ) ] = ( 1 2 ) , and { e , ( 1 2 ) , ( 3 4 ) , ( 1 2 ) ( 3 4 ) } is stable by products and inverse, so K is a subgroup of S 4 , which contains { ( 1 2 ) , ( 1 2 ) ( 3 4 ) } thus H = ⟨ ( 1 2 ) , ( 1 2 ) ( 3 4 ) ⊆ K , so H = K :

⟨ ( 1 2 ) , ( 1 2 ) ( 3 4 ) ⟩ = { e , ( 1 2 ) , ( 3 4 ) , ( 1 2 ) ( 3 4 ) } .

This group of order 4 is not cyclic, because it contains three elements of order 2 . □

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2025-10-23 10:37
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