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!
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.