Express as the product of disjoint cycles and find the order.

a) (1 2 3 5 7)(2 4 7 6)

b) (1 2)(1 3)(1 4)

e) (1 2 3)(3 5 7 9)(1 2 3)^-1

f) (1 2 3 4 5)^3.

Please show steps. Thanks!

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- Apr 24th 2009, 07:47 AMmpryalcycle decomposition 3
Express as the product of disjoint cycles and find the order.

a) (1 2 3 5 7)(2 4 7 6)

b) (1 2)(1 3)(1 4)

e) (1 2 3)(3 5 7 9)(1 2 3)^-1

f) (1 2 3 4 5)^3.

Please show steps. Thanks! - Apr 24th 2009, 08:56 AMThePerfectHacker
If a permutation is expressed as a product of disjoint cycles then the order is the lowest common multiple of all the cycle lengths. This permutation is not a product of disjoint cycles, so we have to turn it into a product of disjoint cycles.

Note (under this permutation):

$\displaystyle 1\mapsto 2\mapsto 4\mapsto 1 \mapsto ... $

$\displaystyle 3\mapsto 5 \mapsto 7 \mapsto 6 \mapsto 3 \mapsto... $

Thus, $\displaystyle (12357)(2476) = (124)(3576)$

This means the order is $\displaystyle \text{lcm}(3,4)=12$

Try the other ones and show your work.