Question

Prove that every element of ${\mathbf{A}}_{\mathbf{n}}$is a product of 3-cycles.

Short answer

Answer:

It is proved that every element of $A_n$is a product of 3-cycles.

Step-by-step solution

Referring to Theorem 7.26 and Theorem 7.29

Theorem 7.26

Every permutation in ${\mathbf{S}}_{\mathbf{n}}$ is a product of (not necessarily disjoint) transpositions.

Theorem 7.29

${\mathbf{A}}_{\mathbf{n}}$ is subgroup of ${\mathbf{S}}_{\mathbf{n}}$of order $\frac{\mathbf{n}\mathbf{!}}{\mathbf{2}}$.

Proving that every element of An is a product of 3-cycles.

As we know each element of $A_n$is a product of even number of transpositions, which implies it is a product of transpositions and each pair is one of the forms $\left(ab\right)\left(cd\right),\left(ab\right)\left(ac\right)$or $\left(ab\right)\left(ab\right)$.

• First considering the formlocalid="1656406065325" $\left(ab\right)\left(cd\right),$as:$\left(ab\right)\left(cd\right)=\left(adb\right)\left(adc\right)$

• Now consideringlocalid="1656406050302" $\left(ab\right)\left(ac\right),$as: $\left(ab\right)\left(ac\right)=\left(acb\right)$

• Now considering $\left(ab\right)\left(ab\right)$as:$\left(ab\right)\left(ab\right)=1$

  $=\left(abc\right)\left(acb\right)$

  
  

As seen from above, in every case, result is the product of 3-cycle.

Hence, it is proved that every element in $A_n$is a product of 3-cycles.