S5 be the permutation group on 5 symbols, then number of element in S5 such that a5= a2a)10b)15c)20d)30Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given symmetric group is S₅.

Then we have to find total no. of elements which satisfied the condition a⁵ = a²

i.e. a⁵ = a² => a³ = e

so, find total no. of elements of order 3. i.e. total no. of cycles of length 3 in S₅

Here r = 3 , n = 5

Answer

Solution:

To find the number of elements in S5 such that a5 = a2a, we need to count the number of permutations in S5 that satisfy this condition.

Permutation Group S5:
The permutation group S5 consists of all possible permutations of the 5 symbols {1, 2, 3, 4, 5}. In other words, it consists of all possible arrangements of these 5 symbols.

Condition a5 = a2a:
The condition a5 = a2a means that the fifth symbol should be equal to the result of applying the permutation a2 followed by the permutation a.

Counting the Number of Permutations:
To count the number of permutations that satisfy the condition a5 = a2a, we can break it down into two steps:

1. Choosing the permutation a2: There are 5 symbols and we need to choose 2 of them to form a2. This can be done in C(5, 2) = 10 ways, where C(n, r) represents the number of combinations of n items taken r at a time.

2. Choosing the permutation a: Once a2 is chosen, the remaining 3 symbols can be permuted in S3. The number of permutations in S3 is 3! = 6.

Therefore, the total number of permutations that satisfy the condition is 10 * 6 = 60.

Answer:
The correct answer is option 'C' (20).

Note: It seems there is an error in the given options. The correct answer should be 60, not 20.

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