Question

3. List the elements of the subgroup of ${\mathit{S}}_7$where

$\mathit{a}=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 2& 7& 6& 5& 1\end{array}\begin{array}{c}7\\ 4\end{array}\right)$.

Short answer

The elements of the subgroup of $S_7$are $\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 2& 7& 6& 5& 1\end{array}\begin{array}{c}7\\ 4\end{array}\right)$,

$$\begin{gathered} \left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 7& 2& 4& 1& 5& 3\end{array}\begin{array}{c}7\\ 6\end{array}\right),\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 4& 2& 6& 3& 5& 7\end{array}\begin{array}{c}7\\ 1\end{array}\right),\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 6& 2& 1& 7& 5& 4\end{array}\begin{array}{c}7\\ 3\end{array}\right), \\ \left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 1& 2& 3& 4& 5& 6\end{array}\begin{array}{c}7\\ 7\end{array}\right) \end{gathered}$$

Step-by-step solution

Cyclic Subgroup

Every element x in a group G, form a subgroup of all integer powers is called a cyclic subgroup of x.

Elements a of S7

The given element of$S_7$ is$a=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 2& 7& 6& 5& 1\end{array}\begin{array}{c}7\\ 4\end{array}\right)$

Cyclic Subgroups a of S7

For element $a\in S_7,a=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 2& 7& 6& 5& 1\end{array}\begin{array}{c}7\\ 4\end{array}\right)$then $a^2=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 7& 2& 4& 1& 5& 3\end{array}\begin{array}{c}7\\ 6\end{array}\right),$

$$\begin{gathered} a^3=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 4& 2& 6& 3& 5& 7\end{array}\begin{array}{c}7\\ 1\end{array}\right),a^4=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 6& 2& 1& 7& 5& 4\end{array}\begin{array}{c}7\\ 3\end{array}\right), \\ {a}^5=\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 1& 2& 3& 4& 5& 6\end{array}\begin{array}{c}7\\ 7\end{array}\right)=I. \end{gathered}$$

Therefore, the elements generated by a are $\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 2& 7& 6& 5& 1\end{array}\begin{array}{c}7\\ 4\end{array}\right),$

$$\begin{gathered} \left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 7& 2& 4& 1& 5& 3\end{array}\begin{array}{c}7\\ 6\end{array}\right),\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 4& 2& 6& 3& 5& 7\end{array}\begin{array}{c}7\\ 1\end{array}\right),\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 6& 2& 1& 7& 5& 4\end{array}\begin{array}{c}7\\ 3\end{array}\right), \\ \left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 1& 2& 3& 4& 5& 6\end{array}\begin{array}{c}7\\ 7\end{array}\right). \end{gathered}$$