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Abstract Algebra

Problem 5.4.6 (Representative of each type of $A_5$ as a commutator in $S_5$.)

Exhibit a representative of each type of A 5 as a commutator in S 5 .

Answers

Proof.

After some trials and errors with GAP, we obtain

( 1 2 3 ) = [ ( 1 3 ) , ( 2 3 ) ] = ( 1 3 ) − 1 ( 2 3 ) − 1 ( 1 3 ) , ( 2 3 ) , ( 1 2 ) ( 3 4 ) = [ ( 1 2 3 ) , ( 1 3 4 ) ] = ( 1 2 3 ) − 1 ( 1 3 4 ) − 1 ( 1 2 3 ) ( 1 3 4 ) , ( 1 2 3 4 5 ) = [ ( 1 2 4 3 ) , ( 1 4 3 5 ) ] = ( 1 2 4 3 ) − 1 ( 1 4 3 5 ) − 1 ( 1 2 4 3 ) ( 1 4 3 5 ) . □
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2026-09-10 10:50
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