Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$. Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$. Determine whether there exists a subgroup of $S_8$ that contains $\alpha$ and $\beta$ and is isomorphic to $D_4$.
 A: HINT: $\alpha^2=\beta^2\ne\mathrm{id}\ne\alpha\beta$
A: $D_4$ has the 8 elements $\{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$, of which exactly two elements have order 4, namely $r$ and $r^3$, where $r^3$ is a power of $r$.  The $\alpha$ and $\beta$ given above each have order 4, but neither is a power of the other.  Hence, any subgroup containing these two elements cannot be isomorphic to $D_4$. 
A: $D_4$ consists of the following: {$()$,$(24)$,$(12)(34)$,$(1234)$,$(13)$,$(13)(24)$,$(1432)$,$(14)(23)$}. The resulting Cayley Table for the group is as follows with the row and column headings corresponding to the same order listed above with $id=()$...$g=(1432),$h$=(14)(23)$
\begin{array}{|c|c|c|c|c|}
\hline
\mathbf{*}&id& b &c& d & e&f&g&h\\
\hline
id &id&b& c & d&e&f&g&h\\
\hline
b & b & id & d&c&f&e&h&g\\
\hline
c & c&g & id & e&d&h&b&f\\
\hline
d & d&h & b & f&c&g&id&e\\
\hline
e & e&f &  g& h&id&b&c&d\\
\hline
f & f&e &  h& g&b&id&d&c\\
\hline
g & g&c &  e& id&h&d&f&b\\
\hline
h & h&d &  f& b&g&c&e&id\\
\hline
\end{array}
The following is a subgroup of $S_8$ that contains $(1234)(5876)$ and $(1537)(2648)$:
{$()$,$(1432)(5678)$, $(13)(24)(57)(68)$,$(1234)(5876)$,$(1735)(2846)$,$(1638)(2745)$,$(1537)(2648)$,   $(1836)(2547)$}
The resulting Cayley Table for the subgroup is as follows with the row and column headings corresponding to the same order as in the subgroup listed above with $id=()$...$g=(1537)(2648)$, $h=(1836)(2247)$
\begin{array}{|c|c|c|c|c|}
\hline
\mathbf{*}&id& b &c& d & e&f&g&h\\
\hline
id &id&b& c & d&e&f&g&h\\
\hline
b & b & c & d&id&f&g&h&e\\
\hline
c & c&d & id & b&g&h&e&f\\
\hline
d & d&id & b & c&h&e&f&g\\
\hline
e & e&h &  g& f&c&b&id&d\\
\hline
f & f&e &  h& g&d&c&b&id\\
\hline
g & g&f &  e& h&id&d&c&b\\
\hline
h & h&g &  f& e&b&id&d&c\\
\hline
\end{array}
See if you can map the elements from one table to the other. I believe you will find they are not isomorphic.
