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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 $D_4$.

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What have you tried ? –  Belgi Feb 11 '13 at 6:07
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Try and see what group those two generate and see if it looks like $D_4$. –  Robert Cardona Feb 11 '13 at 6:08
    
I would start by computing $\beta\alpha\beta^{-1}$ and $\alpha\beta\alpha^{-1}$. It looks promising in a way, but there may be a snag. –  Jyrki Lahtonen Feb 11 '13 at 6:12
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Find the order of these elements. Ask yourself, how many elements of $D_4$ have that order, and how are they related to each other. Then ask whether your two elements are related that way. –  Gerry Myerson Feb 11 '13 at 6:12
    
@JyrkiLahtonen: Does Von Dyck's theorem play a main rule in this problem? –  Babak S. Feb 11 '13 at 6:46

3 Answers 3

HINT: $\alpha^2=\beta^2\ne\mathrm{id}\ne\alpha\beta$

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$D_4$ 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 $D_4$.

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+1 Perfect! You may want to add that $r^{3} = r^{-1}$, and $(r^{-1})^{-1} = r$, so it is enough to check that $\beta$ is not the inverse of $\alpha$. –  Andreas Caranti Mar 10 '13 at 7:50

$D_4$ 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.

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