Math 345/504

3-2-2018

1. Do the following permutation computations, performing any multiplications from right to left. Express your answers in disjoint cycle form.

(a) .

(b) .

(c) .

(d) .

(a) .

(b) .

(c) Since , .

(d) , so .

2. Find the order of the following permutations. Multiply permutations from right to left.

(a) in .

(b) in .

(c) in .

(a) is a 4-cycle, so its order is 4.

(b) The cycles are disjoint, so the order of their product is the least common multiple of their orders. The order is .

(c) The cycles are not disjoint, so I multiply first:

The product is , which has order 3.

* [MATH 504]*

3. Give a specific element of which:

(a) Is a product of two disjoint cycles and has order 3.

(b) Is a product of two disjoint cycles and has order 4.

(c) Has order 2, but does not leave any of fixed.

(a) has order 3 and has order 3. Since the cycles are disjoint, the order of the product is . Therefore, has order 3.

(b) has order 2 and has order 4. Since the cycles are disjoint, the order of the product is . Therefore, has order 4.

(c) , , and all have order 2. Since the cycles are disjoint, the order of the product is . Therefore, has order 2 and does not leave any of fixed.

*I am not afraid of storms, for I am learning to sail my own
ship.* - *Louisa May Alcott*

Copyright 2018 by Bruce Ikenaga