Question

how many elements in S5 and A5 have order 2

Answer 1

Any permutation in has either of the order 1,2,3,4,5,or 6.

S.No.	Cycle notation	Type even/odd	order	In A5 yes/no	count
1	(abcde)	even	5	yes	24
2	(abcd)	odd	4	no
3	(abc) (de)	odd	6	no
4	(abc)	even	3	yes
5	(ab) (cd)	even	2	yes	15
6	(ab)	odd	2	no	10
7	identity	even	1	yes

Out of these the even permutations of order 2 are of the type (ab) (cd) and odd permutations of order 2 are of the type (ab) .

We need to calculate these two types.

The first is even permutations of order 2 are of the type (ab) (cd).

Note that we have 5 letters a,b,c,d,e so to fill the four places we have possibilities, however, out of these 120 possibilities the following 8 types will be assumed as same

Therefore, we need to divide 120 by 8 which gives 15.

Thus in A5 there are 15 elements order 2.

Next the odd permutations of order 2 are of the type (ab), for this simply we choose 2 letters out of 5 in C(5,2)=10 ways.

Thus in S5 there are 15(even)+10(odd)=25 elements order 2.

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