Hw7 Problem 8

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### Problem 8

Find eight elements in $S_6$ that commute with $(12)(34)(56)$. Do they form a subgroup of $S_6$?

**Solution**

In the array below are eight elements in $S_6$ that commute with $(12)(34)(56)$. To better display that the eight elements form a subgroup, observe that each element has an inverse and the subgroup includes the identity. Also, the subgroup is closed. The subgroup whose elements commute with $(12)(34)(56)$ is $S=\{(12)(12),(12),(34),(56),(12)(34),(12)(56),(34)(56),(12)(34)(56)\}$.

(1)\begin{array} {|c|c|c|c|c|c|c|c|} & e & (12) & (34) & (56) & (12)(34) & (12)(56) & (34)(56) & (12)(34)(56) \\ \hline e & e & (12) & (34) & (56) & (12)(34) & (12)(56) & (34)(56) & (12)(34)(56) \\ \hline (12) & (12) & e & (12)(34) & (12)(56) & (34) & (56) & (12)(34)(56) & (34)(56) \\ \hline (34) & (34) & (12)(34) & e & (34)(56) & (12) & (12)(34)(56) & (56) & (12)(56) \\ \hline (56) & (56) & (12)(56) & (34)(56) & e & (12)(34)(56) & (12) & (34) & (12)(34) \\ \hline (12)(34) & (12)(34) & (34) & (12) & (12)(34)(56) & e & (34)(56) & (12)(56) & (56) \\ \hline (12)(56) & (12)(56) & (56) & (12)(34)(56) & (12) & (34) & e & (12)(34) & (34) \\ \hline (34)(56) & (34)(56) & (12)(34)(56) & (12)(56) & (12)(34) & (12)(56) & (12)(34) & e & (12) \\ \hline (12)(34)(56) & (12)(34)(56) & (34)(56) & (12)(56) & (12)(34) & (56) & (34) & (12) & e \\ \end{array}