ABSTRACT ALGEBRA What is the of odd permutations of S6 with

ABSTRACT ALGEBRA: What is the # of odd permutations of S_6 with order 4? Explain.

Solution

We need to factor permutations in disjoint cycles, Now since we are in S6,

we can achieve the order 4 either by using product of any number of disjoint 4-cycles or by using product of any number of disjoint 2 and 4 cycles.

Now we know that even number of disjoint gives odd permutations, So disjoint 4-cycle will give odd permutations.

But a product of disjoint 2 and 4 cycle will give even permutations since odd + odd = even

So we need to calcuate total number of 4-cycles in S6

= 5*4*3 + 4*3*2 + 3*2*1

= 60 + 24 + 6

= 90