Binary tetrahedral group and $rm{SL}_2(mathbb F_3)$
$begingroup$
The binary tetrahedral group $mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ mathbb T = left{ pm 1, pm i, pm j, pm k, dfrac{pm 1 pm i pm j pm k}2 right} subseteq mathbb{H}^times$$
of the multiplicative group of the quaternions (which also gives you a fascinating regular 24-vertex 4-polyhedron).
Alternatively, the binary tetrahedral group can be thought of as the inverse image of $mathfrak A(4)$ through the 2:1 morphism $rm{SU}_2 to rm{SO}_3$ ($mathfrak A(4) subseteq rm{SO}_3$ being the group of direct isometries of the regular tetrahedron).
The property which puzzles me is the following: $mathbb T$ is isomorphic to the matrix group $rm{SL}_2(mathbb F_3)$.
I'm able to give a proof of this result, but not an enlightening one. Essentially, a 24-element group with a normal subgroup isomorphic to the 8-element quaternion group $rm Q_8$ is either isomorphic to $rm Q_8 times rm C_3$ or to the unique nontrivial semidirect product $rm Q_8 rtimes rm C_3$; it's not that hard to prove that both $mathbb T$ and $rm{SL}_2(mathbb F_3)$ satisfy this property and belong to the second case. The only interesting thing about this proof is the description of that normal $rm Q_8$: on the one hand ${pm 1, pm i, pm j, pm k} subset mathbb T$ is the inverse image of the quite exceptional Vierergruppe $V_4 = { rm{id}, (12)(34), (13)(24), (14)(23)} lhd mathfrak A(4)subseteq rm{SO}_3$; on the other hand, the elements of $mathrm{SL}_2(mathbb F_3)$ which are diagonalisable over $mathbb F_9$ — or, equivalently, $pm I_2$ and the matrices whose trace is zero — form a normal subgroup isomorphic to $rm Q_8$ (and it's quite astonishing they do form a subgroup!) whose elements are
$$begin{array}{l} begin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix}, begin{pmatrix} -1 & 0 \ 0 & -1 end{pmatrix}, begin{pmatrix} 0 & 1 \ -1 & 0 end{pmatrix}, begin{pmatrix} 0 & -1 \ 1 & 0 end{pmatrix},\ begin{pmatrix} 1 & 1 \ 1 & -1 end{pmatrix}, begin{pmatrix} 1 & -1 \ -1 & -1 end{pmatrix}, begin{pmatrix} -1 & 1 \ 1 & 1 end{pmatrix}, begin{pmatrix} -1 & -1 \ -1 & 1 end{pmatrix}. end{array}$$
As I said earlier, this proof isn't very satisfying. Hence my question:
Is there an enlightening proof of the isomorphism $mathbb T simeq rm{SL}_2(mathbb F_3)$?
group-theory finite-groups quaternions exceptional-isomorphisms
$endgroup$
|
show 2 more comments
$begingroup$
The binary tetrahedral group $mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ mathbb T = left{ pm 1, pm i, pm j, pm k, dfrac{pm 1 pm i pm j pm k}2 right} subseteq mathbb{H}^times$$
of the multiplicative group of the quaternions (which also gives you a fascinating regular 24-vertex 4-polyhedron).
Alternatively, the binary tetrahedral group can be thought of as the inverse image of $mathfrak A(4)$ through the 2:1 morphism $rm{SU}_2 to rm{SO}_3$ ($mathfrak A(4) subseteq rm{SO}_3$ being the group of direct isometries of the regular tetrahedron).
The property which puzzles me is the following: $mathbb T$ is isomorphic to the matrix group $rm{SL}_2(mathbb F_3)$.
I'm able to give a proof of this result, but not an enlightening one. Essentially, a 24-element group with a normal subgroup isomorphic to the 8-element quaternion group $rm Q_8$ is either isomorphic to $rm Q_8 times rm C_3$ or to the unique nontrivial semidirect product $rm Q_8 rtimes rm C_3$; it's not that hard to prove that both $mathbb T$ and $rm{SL}_2(mathbb F_3)$ satisfy this property and belong to the second case. The only interesting thing about this proof is the description of that normal $rm Q_8$: on the one hand ${pm 1, pm i, pm j, pm k} subset mathbb T$ is the inverse image of the quite exceptional Vierergruppe $V_4 = { rm{id}, (12)(34), (13)(24), (14)(23)} lhd mathfrak A(4)subseteq rm{SO}_3$; on the other hand, the elements of $mathrm{SL}_2(mathbb F_3)$ which are diagonalisable over $mathbb F_9$ — or, equivalently, $pm I_2$ and the matrices whose trace is zero — form a normal subgroup isomorphic to $rm Q_8$ (and it's quite astonishing they do form a subgroup!) whose elements are
$$begin{array}{l} begin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix}, begin{pmatrix} -1 & 0 \ 0 & -1 end{pmatrix}, begin{pmatrix} 0 & 1 \ -1 & 0 end{pmatrix}, begin{pmatrix} 0 & -1 \ 1 & 0 end{pmatrix},\ begin{pmatrix} 1 & 1 \ 1 & -1 end{pmatrix}, begin{pmatrix} 1 & -1 \ -1 & -1 end{pmatrix}, begin{pmatrix} -1 & 1 \ 1 & 1 end{pmatrix}, begin{pmatrix} -1 & -1 \ -1 & 1 end{pmatrix}. end{array}$$
As I said earlier, this proof isn't very satisfying. Hence my question:
Is there an enlightening proof of the isomorphism $mathbb T simeq rm{SL}_2(mathbb F_3)$?
group-theory finite-groups quaternions exceptional-isomorphisms
$endgroup$
$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
$endgroup$
– Qiaochu Yuan
Apr 24 '13 at 17:41
2
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:35
2
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
3
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
1
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59
|
show 2 more comments
$begingroup$
The binary tetrahedral group $mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ mathbb T = left{ pm 1, pm i, pm j, pm k, dfrac{pm 1 pm i pm j pm k}2 right} subseteq mathbb{H}^times$$
of the multiplicative group of the quaternions (which also gives you a fascinating regular 24-vertex 4-polyhedron).
Alternatively, the binary tetrahedral group can be thought of as the inverse image of $mathfrak A(4)$ through the 2:1 morphism $rm{SU}_2 to rm{SO}_3$ ($mathfrak A(4) subseteq rm{SO}_3$ being the group of direct isometries of the regular tetrahedron).
The property which puzzles me is the following: $mathbb T$ is isomorphic to the matrix group $rm{SL}_2(mathbb F_3)$.
I'm able to give a proof of this result, but not an enlightening one. Essentially, a 24-element group with a normal subgroup isomorphic to the 8-element quaternion group $rm Q_8$ is either isomorphic to $rm Q_8 times rm C_3$ or to the unique nontrivial semidirect product $rm Q_8 rtimes rm C_3$; it's not that hard to prove that both $mathbb T$ and $rm{SL}_2(mathbb F_3)$ satisfy this property and belong to the second case. The only interesting thing about this proof is the description of that normal $rm Q_8$: on the one hand ${pm 1, pm i, pm j, pm k} subset mathbb T$ is the inverse image of the quite exceptional Vierergruppe $V_4 = { rm{id}, (12)(34), (13)(24), (14)(23)} lhd mathfrak A(4)subseteq rm{SO}_3$; on the other hand, the elements of $mathrm{SL}_2(mathbb F_3)$ which are diagonalisable over $mathbb F_9$ — or, equivalently, $pm I_2$ and the matrices whose trace is zero — form a normal subgroup isomorphic to $rm Q_8$ (and it's quite astonishing they do form a subgroup!) whose elements are
$$begin{array}{l} begin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix}, begin{pmatrix} -1 & 0 \ 0 & -1 end{pmatrix}, begin{pmatrix} 0 & 1 \ -1 & 0 end{pmatrix}, begin{pmatrix} 0 & -1 \ 1 & 0 end{pmatrix},\ begin{pmatrix} 1 & 1 \ 1 & -1 end{pmatrix}, begin{pmatrix} 1 & -1 \ -1 & -1 end{pmatrix}, begin{pmatrix} -1 & 1 \ 1 & 1 end{pmatrix}, begin{pmatrix} -1 & -1 \ -1 & 1 end{pmatrix}. end{array}$$
As I said earlier, this proof isn't very satisfying. Hence my question:
Is there an enlightening proof of the isomorphism $mathbb T simeq rm{SL}_2(mathbb F_3)$?
group-theory finite-groups quaternions exceptional-isomorphisms
$endgroup$
The binary tetrahedral group $mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ mathbb T = left{ pm 1, pm i, pm j, pm k, dfrac{pm 1 pm i pm j pm k}2 right} subseteq mathbb{H}^times$$
of the multiplicative group of the quaternions (which also gives you a fascinating regular 24-vertex 4-polyhedron).
Alternatively, the binary tetrahedral group can be thought of as the inverse image of $mathfrak A(4)$ through the 2:1 morphism $rm{SU}_2 to rm{SO}_3$ ($mathfrak A(4) subseteq rm{SO}_3$ being the group of direct isometries of the regular tetrahedron).
The property which puzzles me is the following: $mathbb T$ is isomorphic to the matrix group $rm{SL}_2(mathbb F_3)$.
I'm able to give a proof of this result, but not an enlightening one. Essentially, a 24-element group with a normal subgroup isomorphic to the 8-element quaternion group $rm Q_8$ is either isomorphic to $rm Q_8 times rm C_3$ or to the unique nontrivial semidirect product $rm Q_8 rtimes rm C_3$; it's not that hard to prove that both $mathbb T$ and $rm{SL}_2(mathbb F_3)$ satisfy this property and belong to the second case. The only interesting thing about this proof is the description of that normal $rm Q_8$: on the one hand ${pm 1, pm i, pm j, pm k} subset mathbb T$ is the inverse image of the quite exceptional Vierergruppe $V_4 = { rm{id}, (12)(34), (13)(24), (14)(23)} lhd mathfrak A(4)subseteq rm{SO}_3$; on the other hand, the elements of $mathrm{SL}_2(mathbb F_3)$ which are diagonalisable over $mathbb F_9$ — or, equivalently, $pm I_2$ and the matrices whose trace is zero — form a normal subgroup isomorphic to $rm Q_8$ (and it's quite astonishing they do form a subgroup!) whose elements are
$$begin{array}{l} begin{pmatrix} 1 & 0 \ 0 & 1 end{pmatrix}, begin{pmatrix} -1 & 0 \ 0 & -1 end{pmatrix}, begin{pmatrix} 0 & 1 \ -1 & 0 end{pmatrix}, begin{pmatrix} 0 & -1 \ 1 & 0 end{pmatrix},\ begin{pmatrix} 1 & 1 \ 1 & -1 end{pmatrix}, begin{pmatrix} 1 & -1 \ -1 & -1 end{pmatrix}, begin{pmatrix} -1 & 1 \ 1 & 1 end{pmatrix}, begin{pmatrix} -1 & -1 \ -1 & 1 end{pmatrix}. end{array}$$
As I said earlier, this proof isn't very satisfying. Hence my question:
Is there an enlightening proof of the isomorphism $mathbb T simeq rm{SL}_2(mathbb F_3)$?
group-theory finite-groups quaternions exceptional-isomorphisms
group-theory finite-groups quaternions exceptional-isomorphisms
edited May 24 '13 at 11:44
Grigory M
13.6k357104
13.6k357104
asked Apr 24 '13 at 14:17
PseudoNeoPseudoNeo
6,5492053
6,5492053
$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
$endgroup$
– Qiaochu Yuan
Apr 24 '13 at 17:41
2
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:35
2
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
3
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
1
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59
|
show 2 more comments
$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
$endgroup$
– Qiaochu Yuan
Apr 24 '13 at 17:41
2
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:35
2
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
3
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
1
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59
$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
$endgroup$
– Qiaochu Yuan
Apr 24 '13 at 17:41
$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
$endgroup$
– Qiaochu Yuan
Apr 24 '13 at 17:41
2
2
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:35
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:35
2
2
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
3
3
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
1
1
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59
|
show 2 more comments
3 Answers
3
active
oldest
votes
$begingroup$
Let's start with the ring $H$ of Hurwitz quaternions (quaternions s.t. either all coefficients are integers or all coefficients are half-integers). The ring $Hotimesmathbb F_p$ is a quaternion algebra over a finite field, i.e. a matrix algebra $Mat_{2times 2}(mathbb F_p)$ (with determinant as the norm).
Now the binary tetrahedral group is the group of norm $1$ elements in $H$ and it maps injectively (for $p>2$) to the group of norm $1$ elements in $Hotimesmathbb F_p$, i.e. to $SL_2(mathbb F_p)$. In particular, for $p=3$ we get an isomorphism $mathbb Tto SL_2(mathbb F_3)$.
(Ref.: TWF 198.)
$endgroup$
add a comment |
$begingroup$
We know that $mathbb T= <x,y,z| x^2=y^3=z^3=xyz>$. We also know that $ord(xyz)=2$, $mathbb{T}/<xyz> simeq A_4$ and $mathbb{T}$ has no subgroup of order 12.
In one of my problem sheets, we are intended to give a guided proof of the fact that $mathbb T simeq SL_2( mathbb F_3)$ by showing that the map $phi: mathbb T to SL_2(mathbb F_3)$ that sends $$x longrightarrow left( begin{array}{cc} 0 & 2 \ 1 & 0 end{array} right), y longrightarrow left( begin{array}{cc} 2 & 0 \ 1 & 2 end{array} right) text{ and } z longrightarrow left( begin{array}{cc} 0 & 1 \ 2 & 1 end{array} right)$$ extends to an isomorphism between $mathbb T$ and $SL_2( mathbb{F}_3)$. This map is clearly a homomorphism, because it is easy to see that the images of the generators respect the relations.
We can just count the elements of $SL_2(mathbb F_3)$ to see that $|SL_2(mathbb F_3)|= | mathbb{T}|=24$, so the only thing that we still have to show is either that $phi$ is injective or surjective. Anybody can think of an easy way to prove this? I think this might lead to an easy proof of the result, without even making use of the notion of semidirect products.
$endgroup$
add a comment |
$begingroup$
Maybe the confusion arrises from the fact that $mathbb{T}$
has no subgroup of order $12$.
Rather it has the quaternion group
$Q={pm1,pm i,pm j, pm k}$
as a subgroup of order $8$.
I think a good way to grasp this intuitively is to see it in the context of the well known action of $mathbb{H}^1$
on the pure quaternions $mathbb{H}_0simeqmathbb{R}^3$
by rotation via $q(p)=qpoverline{q}$,
which gives a double cover of $mathrm{SO}(3)$.
In particular,
$mathbb{H}^1/{pm1}congmathrm{SO}(3)$.
We can use this to realize the action of $mathbb{T}$
on a tetrahedron explicitly, as follows.
Let $Tsubsetmathbb{H}_0$
be the tetrahedron with vertices
$$i+j+k,quad i-j-k,quad -i+j-k,quad -i-j+k.$$
An element of $Q$
acts on this via rotation by $pi$
about a line through the midpoints of opposite sides of $T$.
An element of $mathbb{T}-Q$
acts on this by rotation by $2pi/3$
about a line through a vertex and the midpoint of its opposite face.
Any pair of elements of $mathbb{T}$
differing only by sign give the same action.
Now, by passing from $Q$
to $mathbb{T}$
we have tripled the size of the group,
and included everything necessary to get the full tetrahedral group,
but with redundancy.
In particular, the kernel of the action of $mathbb{T}$ on $T$ is ${pm1}$.
So $mathbb{T}$
gives a double cover of the tetrahedral group.
As you mentioned, there's a straightforward hands-on way to see that the tetrahedral group $mathfrak{U}(4)$
is isomorphic to $mathrm{PSL}(2,mathbb{F}_3)$.
So if we pull back to the full group,
we've got $mathbb{T}congmathrm{SL}_2(mathbb{F}_3)$.
This way of thinking about it is not my own invention. It's from a preprint I have of the book Quaternion Algebras by John Voight, which is expected to go into print very soon and which I highly recommend.
$endgroup$
add a comment |
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3 Answers
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active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let's start with the ring $H$ of Hurwitz quaternions (quaternions s.t. either all coefficients are integers or all coefficients are half-integers). The ring $Hotimesmathbb F_p$ is a quaternion algebra over a finite field, i.e. a matrix algebra $Mat_{2times 2}(mathbb F_p)$ (with determinant as the norm).
Now the binary tetrahedral group is the group of norm $1$ elements in $H$ and it maps injectively (for $p>2$) to the group of norm $1$ elements in $Hotimesmathbb F_p$, i.e. to $SL_2(mathbb F_p)$. In particular, for $p=3$ we get an isomorphism $mathbb Tto SL_2(mathbb F_3)$.
(Ref.: TWF 198.)
$endgroup$
add a comment |
$begingroup$
Let's start with the ring $H$ of Hurwitz quaternions (quaternions s.t. either all coefficients are integers or all coefficients are half-integers). The ring $Hotimesmathbb F_p$ is a quaternion algebra over a finite field, i.e. a matrix algebra $Mat_{2times 2}(mathbb F_p)$ (with determinant as the norm).
Now the binary tetrahedral group is the group of norm $1$ elements in $H$ and it maps injectively (for $p>2$) to the group of norm $1$ elements in $Hotimesmathbb F_p$, i.e. to $SL_2(mathbb F_p)$. In particular, for $p=3$ we get an isomorphism $mathbb Tto SL_2(mathbb F_3)$.
(Ref.: TWF 198.)
$endgroup$
add a comment |
$begingroup$
Let's start with the ring $H$ of Hurwitz quaternions (quaternions s.t. either all coefficients are integers or all coefficients are half-integers). The ring $Hotimesmathbb F_p$ is a quaternion algebra over a finite field, i.e. a matrix algebra $Mat_{2times 2}(mathbb F_p)$ (with determinant as the norm).
Now the binary tetrahedral group is the group of norm $1$ elements in $H$ and it maps injectively (for $p>2$) to the group of norm $1$ elements in $Hotimesmathbb F_p$, i.e. to $SL_2(mathbb F_p)$. In particular, for $p=3$ we get an isomorphism $mathbb Tto SL_2(mathbb F_3)$.
(Ref.: TWF 198.)
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Let's start with the ring $H$ of Hurwitz quaternions (quaternions s.t. either all coefficients are integers or all coefficients are half-integers). The ring $Hotimesmathbb F_p$ is a quaternion algebra over a finite field, i.e. a matrix algebra $Mat_{2times 2}(mathbb F_p)$ (with determinant as the norm).
Now the binary tetrahedral group is the group of norm $1$ elements in $H$ and it maps injectively (for $p>2$) to the group of norm $1$ elements in $Hotimesmathbb F_p$, i.e. to $SL_2(mathbb F_p)$. In particular, for $p=3$ we get an isomorphism $mathbb Tto SL_2(mathbb F_3)$.
(Ref.: TWF 198.)
edited Dec 19 '18 at 4:17
the_fox
2,90031538
2,90031538
answered May 24 '13 at 11:36
Grigory MGrigory M
13.6k357104
13.6k357104
add a comment |
add a comment |
$begingroup$
We know that $mathbb T= <x,y,z| x^2=y^3=z^3=xyz>$. We also know that $ord(xyz)=2$, $mathbb{T}/<xyz> simeq A_4$ and $mathbb{T}$ has no subgroup of order 12.
In one of my problem sheets, we are intended to give a guided proof of the fact that $mathbb T simeq SL_2( mathbb F_3)$ by showing that the map $phi: mathbb T to SL_2(mathbb F_3)$ that sends $$x longrightarrow left( begin{array}{cc} 0 & 2 \ 1 & 0 end{array} right), y longrightarrow left( begin{array}{cc} 2 & 0 \ 1 & 2 end{array} right) text{ and } z longrightarrow left( begin{array}{cc} 0 & 1 \ 2 & 1 end{array} right)$$ extends to an isomorphism between $mathbb T$ and $SL_2( mathbb{F}_3)$. This map is clearly a homomorphism, because it is easy to see that the images of the generators respect the relations.
We can just count the elements of $SL_2(mathbb F_3)$ to see that $|SL_2(mathbb F_3)|= | mathbb{T}|=24$, so the only thing that we still have to show is either that $phi$ is injective or surjective. Anybody can think of an easy way to prove this? I think this might lead to an easy proof of the result, without even making use of the notion of semidirect products.
$endgroup$
add a comment |
$begingroup$
We know that $mathbb T= <x,y,z| x^2=y^3=z^3=xyz>$. We also know that $ord(xyz)=2$, $mathbb{T}/<xyz> simeq A_4$ and $mathbb{T}$ has no subgroup of order 12.
In one of my problem sheets, we are intended to give a guided proof of the fact that $mathbb T simeq SL_2( mathbb F_3)$ by showing that the map $phi: mathbb T to SL_2(mathbb F_3)$ that sends $$x longrightarrow left( begin{array}{cc} 0 & 2 \ 1 & 0 end{array} right), y longrightarrow left( begin{array}{cc} 2 & 0 \ 1 & 2 end{array} right) text{ and } z longrightarrow left( begin{array}{cc} 0 & 1 \ 2 & 1 end{array} right)$$ extends to an isomorphism between $mathbb T$ and $SL_2( mathbb{F}_3)$. This map is clearly a homomorphism, because it is easy to see that the images of the generators respect the relations.
We can just count the elements of $SL_2(mathbb F_3)$ to see that $|SL_2(mathbb F_3)|= | mathbb{T}|=24$, so the only thing that we still have to show is either that $phi$ is injective or surjective. Anybody can think of an easy way to prove this? I think this might lead to an easy proof of the result, without even making use of the notion of semidirect products.
$endgroup$
add a comment |
$begingroup$
We know that $mathbb T= <x,y,z| x^2=y^3=z^3=xyz>$. We also know that $ord(xyz)=2$, $mathbb{T}/<xyz> simeq A_4$ and $mathbb{T}$ has no subgroup of order 12.
In one of my problem sheets, we are intended to give a guided proof of the fact that $mathbb T simeq SL_2( mathbb F_3)$ by showing that the map $phi: mathbb T to SL_2(mathbb F_3)$ that sends $$x longrightarrow left( begin{array}{cc} 0 & 2 \ 1 & 0 end{array} right), y longrightarrow left( begin{array}{cc} 2 & 0 \ 1 & 2 end{array} right) text{ and } z longrightarrow left( begin{array}{cc} 0 & 1 \ 2 & 1 end{array} right)$$ extends to an isomorphism between $mathbb T$ and $SL_2( mathbb{F}_3)$. This map is clearly a homomorphism, because it is easy to see that the images of the generators respect the relations.
We can just count the elements of $SL_2(mathbb F_3)$ to see that $|SL_2(mathbb F_3)|= | mathbb{T}|=24$, so the only thing that we still have to show is either that $phi$ is injective or surjective. Anybody can think of an easy way to prove this? I think this might lead to an easy proof of the result, without even making use of the notion of semidirect products.
$endgroup$
We know that $mathbb T= <x,y,z| x^2=y^3=z^3=xyz>$. We also know that $ord(xyz)=2$, $mathbb{T}/<xyz> simeq A_4$ and $mathbb{T}$ has no subgroup of order 12.
In one of my problem sheets, we are intended to give a guided proof of the fact that $mathbb T simeq SL_2( mathbb F_3)$ by showing that the map $phi: mathbb T to SL_2(mathbb F_3)$ that sends $$x longrightarrow left( begin{array}{cc} 0 & 2 \ 1 & 0 end{array} right), y longrightarrow left( begin{array}{cc} 2 & 0 \ 1 & 2 end{array} right) text{ and } z longrightarrow left( begin{array}{cc} 0 & 1 \ 2 & 1 end{array} right)$$ extends to an isomorphism between $mathbb T$ and $SL_2( mathbb{F}_3)$. This map is clearly a homomorphism, because it is easy to see that the images of the generators respect the relations.
We can just count the elements of $SL_2(mathbb F_3)$ to see that $|SL_2(mathbb F_3)|= | mathbb{T}|=24$, so the only thing that we still have to show is either that $phi$ is injective or surjective. Anybody can think of an easy way to prove this? I think this might lead to an easy proof of the result, without even making use of the notion of semidirect products.
answered Mar 1 '14 at 9:40
George TurcasGeorge Turcas
513
513
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$begingroup$
Maybe the confusion arrises from the fact that $mathbb{T}$
has no subgroup of order $12$.
Rather it has the quaternion group
$Q={pm1,pm i,pm j, pm k}$
as a subgroup of order $8$.
I think a good way to grasp this intuitively is to see it in the context of the well known action of $mathbb{H}^1$
on the pure quaternions $mathbb{H}_0simeqmathbb{R}^3$
by rotation via $q(p)=qpoverline{q}$,
which gives a double cover of $mathrm{SO}(3)$.
In particular,
$mathbb{H}^1/{pm1}congmathrm{SO}(3)$.
We can use this to realize the action of $mathbb{T}$
on a tetrahedron explicitly, as follows.
Let $Tsubsetmathbb{H}_0$
be the tetrahedron with vertices
$$i+j+k,quad i-j-k,quad -i+j-k,quad -i-j+k.$$
An element of $Q$
acts on this via rotation by $pi$
about a line through the midpoints of opposite sides of $T$.
An element of $mathbb{T}-Q$
acts on this by rotation by $2pi/3$
about a line through a vertex and the midpoint of its opposite face.
Any pair of elements of $mathbb{T}$
differing only by sign give the same action.
Now, by passing from $Q$
to $mathbb{T}$
we have tripled the size of the group,
and included everything necessary to get the full tetrahedral group,
but with redundancy.
In particular, the kernel of the action of $mathbb{T}$ on $T$ is ${pm1}$.
So $mathbb{T}$
gives a double cover of the tetrahedral group.
As you mentioned, there's a straightforward hands-on way to see that the tetrahedral group $mathfrak{U}(4)$
is isomorphic to $mathrm{PSL}(2,mathbb{F}_3)$.
So if we pull back to the full group,
we've got $mathbb{T}congmathrm{SL}_2(mathbb{F}_3)$.
This way of thinking about it is not my own invention. It's from a preprint I have of the book Quaternion Algebras by John Voight, which is expected to go into print very soon and which I highly recommend.
$endgroup$
add a comment |
$begingroup$
Maybe the confusion arrises from the fact that $mathbb{T}$
has no subgroup of order $12$.
Rather it has the quaternion group
$Q={pm1,pm i,pm j, pm k}$
as a subgroup of order $8$.
I think a good way to grasp this intuitively is to see it in the context of the well known action of $mathbb{H}^1$
on the pure quaternions $mathbb{H}_0simeqmathbb{R}^3$
by rotation via $q(p)=qpoverline{q}$,
which gives a double cover of $mathrm{SO}(3)$.
In particular,
$mathbb{H}^1/{pm1}congmathrm{SO}(3)$.
We can use this to realize the action of $mathbb{T}$
on a tetrahedron explicitly, as follows.
Let $Tsubsetmathbb{H}_0$
be the tetrahedron with vertices
$$i+j+k,quad i-j-k,quad -i+j-k,quad -i-j+k.$$
An element of $Q$
acts on this via rotation by $pi$
about a line through the midpoints of opposite sides of $T$.
An element of $mathbb{T}-Q$
acts on this by rotation by $2pi/3$
about a line through a vertex and the midpoint of its opposite face.
Any pair of elements of $mathbb{T}$
differing only by sign give the same action.
Now, by passing from $Q$
to $mathbb{T}$
we have tripled the size of the group,
and included everything necessary to get the full tetrahedral group,
but with redundancy.
In particular, the kernel of the action of $mathbb{T}$ on $T$ is ${pm1}$.
So $mathbb{T}$
gives a double cover of the tetrahedral group.
As you mentioned, there's a straightforward hands-on way to see that the tetrahedral group $mathfrak{U}(4)$
is isomorphic to $mathrm{PSL}(2,mathbb{F}_3)$.
So if we pull back to the full group,
we've got $mathbb{T}congmathrm{SL}_2(mathbb{F}_3)$.
This way of thinking about it is not my own invention. It's from a preprint I have of the book Quaternion Algebras by John Voight, which is expected to go into print very soon and which I highly recommend.
$endgroup$
add a comment |
$begingroup$
Maybe the confusion arrises from the fact that $mathbb{T}$
has no subgroup of order $12$.
Rather it has the quaternion group
$Q={pm1,pm i,pm j, pm k}$
as a subgroup of order $8$.
I think a good way to grasp this intuitively is to see it in the context of the well known action of $mathbb{H}^1$
on the pure quaternions $mathbb{H}_0simeqmathbb{R}^3$
by rotation via $q(p)=qpoverline{q}$,
which gives a double cover of $mathrm{SO}(3)$.
In particular,
$mathbb{H}^1/{pm1}congmathrm{SO}(3)$.
We can use this to realize the action of $mathbb{T}$
on a tetrahedron explicitly, as follows.
Let $Tsubsetmathbb{H}_0$
be the tetrahedron with vertices
$$i+j+k,quad i-j-k,quad -i+j-k,quad -i-j+k.$$
An element of $Q$
acts on this via rotation by $pi$
about a line through the midpoints of opposite sides of $T$.
An element of $mathbb{T}-Q$
acts on this by rotation by $2pi/3$
about a line through a vertex and the midpoint of its opposite face.
Any pair of elements of $mathbb{T}$
differing only by sign give the same action.
Now, by passing from $Q$
to $mathbb{T}$
we have tripled the size of the group,
and included everything necessary to get the full tetrahedral group,
but with redundancy.
In particular, the kernel of the action of $mathbb{T}$ on $T$ is ${pm1}$.
So $mathbb{T}$
gives a double cover of the tetrahedral group.
As you mentioned, there's a straightforward hands-on way to see that the tetrahedral group $mathfrak{U}(4)$
is isomorphic to $mathrm{PSL}(2,mathbb{F}_3)$.
So if we pull back to the full group,
we've got $mathbb{T}congmathrm{SL}_2(mathbb{F}_3)$.
This way of thinking about it is not my own invention. It's from a preprint I have of the book Quaternion Algebras by John Voight, which is expected to go into print very soon and which I highly recommend.
$endgroup$
Maybe the confusion arrises from the fact that $mathbb{T}$
has no subgroup of order $12$.
Rather it has the quaternion group
$Q={pm1,pm i,pm j, pm k}$
as a subgroup of order $8$.
I think a good way to grasp this intuitively is to see it in the context of the well known action of $mathbb{H}^1$
on the pure quaternions $mathbb{H}_0simeqmathbb{R}^3$
by rotation via $q(p)=qpoverline{q}$,
which gives a double cover of $mathrm{SO}(3)$.
In particular,
$mathbb{H}^1/{pm1}congmathrm{SO}(3)$.
We can use this to realize the action of $mathbb{T}$
on a tetrahedron explicitly, as follows.
Let $Tsubsetmathbb{H}_0$
be the tetrahedron with vertices
$$i+j+k,quad i-j-k,quad -i+j-k,quad -i-j+k.$$
An element of $Q$
acts on this via rotation by $pi$
about a line through the midpoints of opposite sides of $T$.
An element of $mathbb{T}-Q$
acts on this by rotation by $2pi/3$
about a line through a vertex and the midpoint of its opposite face.
Any pair of elements of $mathbb{T}$
differing only by sign give the same action.
Now, by passing from $Q$
to $mathbb{T}$
we have tripled the size of the group,
and included everything necessary to get the full tetrahedral group,
but with redundancy.
In particular, the kernel of the action of $mathbb{T}$ on $T$ is ${pm1}$.
So $mathbb{T}$
gives a double cover of the tetrahedral group.
As you mentioned, there's a straightforward hands-on way to see that the tetrahedral group $mathfrak{U}(4)$
is isomorphic to $mathrm{PSL}(2,mathbb{F}_3)$.
So if we pull back to the full group,
we've got $mathbb{T}congmathrm{SL}_2(mathbb{F}_3)$.
This way of thinking about it is not my own invention. It's from a preprint I have of the book Quaternion Algebras by John Voight, which is expected to go into print very soon and which I highly recommend.
edited Dec 18 '18 at 21:08
answered Feb 22 '17 at 1:10
j0equ1nnj0equ1nn
1,5571023
1,5571023
add a comment |
add a comment |
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$begingroup$
I suspect that the binary tetrahedral group is conjugate to a subgroup of $text{SL}_2(mathbb{Z})$. If so, it's known that if $G$ is a finite subgroup of $text{GL}_n(mathbb{Z})$ then $G$ injects into $text{GL}_n(mathbb{F}_3)$, and that provides a pretty straightforward isomorphism.
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– Qiaochu Yuan
Apr 24 '13 at 17:41
2
$begingroup$
The isomorphism $mathfrak A(4) simeq rm{PSL}_2(mathbb F_3)$ is quite easy to see: the action of $rm{PSL}_2(mathbb F_3)$ on the 4-element projective line $rm P^1(mathbb F_3)$ provides an injective morphism $rm{PSL}_2(mathbb F_3) to mathfrak S(4)$ which gives easily the desired isomorphism. What is unclear to me is why the double cover gives precisely $rm{SL}_2(mathbb F_3)$. Maybe there are direct cohomological arguments?
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– PseudoNeo
Apr 24 '13 at 19:35
2
$begingroup$
@QiaochuYuan: I doubt it: finite subgroups of $rm{GL}_2(mathbb Z)$ preserve a scalar product, so they are cyclic or dihedral.
$endgroup$
– PseudoNeo
Apr 24 '13 at 19:38
3
$begingroup$
Well, this is probably not as geometric as you want, but $SL(2,3)$ is the only nontrivial "double cover" (central extension of order 2) over $PSL(2,3)$.
$endgroup$
– user641
Apr 24 '13 at 23:24
1
$begingroup$
There is a 2-dimensional complex representation of $Q_8$, but it cannot be written over the rationals, so $Q_8$ does not embed in ${rm GL}_2({mathbb Z})$. (But ${rm SL}_2(3)$, which contains $Q_8$, does embed in ${rm SL}_2(p)$ for all odd primes $p$.)
$endgroup$
– Derek Holt
Apr 25 '13 at 8:59