Question

What is the order of each group:

(d).$S_5$

Short answer

The order of group $S_5$ is $120$.

Step-by-step solution

Permutation

Let$S$be a non-empty set. A permutation on $S$is defined as bijective mapping

$f:S\to S$. If, $S=\left\{a_1,a_2,.......,a_n\right\}$.

Then, the number of bijection from$S$onto $S$is $n!$.

Let one of the bijection be $f$which maps $a_i$to $f\left(a_i\right)$. This is denoted by the symbol,

$\left(\begin{array}{cccc}{a}_1& a_2& ...& a_n\\ f\left(a_1\right)& f\left(a_2\right)& ...& f\left(a_n\right)\\ & & & \end{array}\right)$

Order of a group

The order of a group $G$ is defined as the number of elements in the group.

Conclusion

Since $S_5$ is a group of permutations of $5$elements,

So, the order of $S_5$ is given by $5!=120$