Question

Is the set ${\mathit{B}}_{\mathbf{n}}$ of odd permutation in${\mathit{S}}_{\mathbf{n}}$ a group? Justify your answer.

Short answer

The set $B_n$of odd permutations in $S_n$is not a group.

Step-by-step solution

Conditions for odd permutation

If set $B_n$of odd permutation in $S_n$is a group then, it must satisfy two conditions:

1. It must have an odd identity permutation, and
2. It should be closed under the composition.

Now, we have to verify both the conditions.

Proving that Set Bn of odd permutation in Sn is not a group

Since the identity permutation in$S_n$is always even, the Bn does not have any odd identity element.

Similarly, we can also show that $B_n$is not closed under the composition of transposition.

For example, let’s take two odd permutations ( 12 ) and ( 23 ).

Its transposition can be given as $\left(12\right)\left(23\right)=\left(123\right)$,

which is even because the addition of two odd numbers is always an even number.

Since both the conditions 1 and 2 do not hold for $B_n$. Hence, set of odd permutations in $S_n$is not a group.