Permutation groups: $S_4$ and $D_4$.
$begingroup$
Question. Determine the subgroup of $S_4$ generated by $sigma=(1 2 3 4)$ e $tau = (2 4)$. Show that $left<sigma, tauright> <S_4$ is isomorphic to the group of square symmetries.
Attempt to solve. I found that, $left<sigma, tauright>={e, sigma, sigma^2, sigma^3, tau, sigma circ tau, tau circ sigma, sigma^2 circ tau}$. We have that, the group of symmetries of square $D_4$ has $8$ elements. Consider the application
$$f : left<sigma, tauright> to D_4$$
I need help defining this application and showing that there is an isomorphism.
group-theory permutations symmetric-groups group-isomorphism
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
$endgroup$
add a comment |
$begingroup$
Question. Determine the subgroup of $S_4$ generated by $sigma=(1 2 3 4)$ e $tau = (2 4)$. Show that $left<sigma, tauright> <S_4$ is isomorphic to the group of square symmetries.
Attempt to solve. I found that, $left<sigma, tauright>={e, sigma, sigma^2, sigma^3, tau, sigma circ tau, tau circ sigma, sigma^2 circ tau}$. We have that, the group of symmetries of square $D_4$ has $8$ elements. Consider the application
$$f : left<sigma, tauright> to D_4$$
I need help defining this application and showing that there is an isomorphism.
group-theory permutations symmetric-groups group-isomorphism
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
$endgroup$
2
$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
1
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
1
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18
add a comment |
$begingroup$
Question. Determine the subgroup of $S_4$ generated by $sigma=(1 2 3 4)$ e $tau = (2 4)$. Show that $left<sigma, tauright> <S_4$ is isomorphic to the group of square symmetries.
Attempt to solve. I found that, $left<sigma, tauright>={e, sigma, sigma^2, sigma^3, tau, sigma circ tau, tau circ sigma, sigma^2 circ tau}$. We have that, the group of symmetries of square $D_4$ has $8$ elements. Consider the application
$$f : left<sigma, tauright> to D_4$$
I need help defining this application and showing that there is an isomorphism.
group-theory permutations symmetric-groups group-isomorphism
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
$endgroup$
Question. Determine the subgroup of $S_4$ generated by $sigma=(1 2 3 4)$ e $tau = (2 4)$. Show that $left<sigma, tauright> <S_4$ is isomorphic to the group of square symmetries.
Attempt to solve. I found that, $left<sigma, tauright>={e, sigma, sigma^2, sigma^3, tau, sigma circ tau, tau circ sigma, sigma^2 circ tau}$. We have that, the group of symmetries of square $D_4$ has $8$ elements. Consider the application
$$f : left<sigma, tauright> to D_4$$
I need help defining this application and showing that there is an isomorphism.
group-theory permutations symmetric-groups group-isomorphism
group-theory permutations symmetric-groups group-isomorphism
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
asked Dec 12 '18 at 2:26
wavilson ferreirawavilson ferreira
517
517
2
$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
1
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
1
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18
add a comment |
2
$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
1
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
1
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18
2
2
$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
1
1
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
1
1
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18
add a comment |
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$begingroup$
You have already done all of the hard work. Find elements in $D_4$ that act like $sigma$ and $tau$ and that will give your isomorphism.
$endgroup$
– Randall
Dec 12 '18 at 2:27
1
$begingroup$
@wavilsonferreira: Map generators of $langlesigma,taurangle$ to generators of $D_4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 2:31
$begingroup$
$D_4={e,r_1,r_2,r_3,s_1,s_2,s_3,s_4}$ where $r_i$ are the rotations and $s_i$ the reflections, I know that $left<sigma,tauright>$ and $D_4$ coincide as follows $sigma=r_1,sigma^2=r_2,...,sigma^2 circ tau=s_4$. @YadatiKiran how can I do this, considering that the $D_4$ generators are $r_1$ and $s_4$, if those generators are wrong, could you tell me the correct ones?
$endgroup$
– wavilson ferreira
Dec 12 '18 at 2:49
1
$begingroup$
@wavilsonferreira: Consider mapping $sigmamapsto a$ and $taumapsto b$ where $a,bin D_4$ such that $b^2=e=a^4$.
$endgroup$
– Yadati Kiran
Dec 12 '18 at 4:18