Our aim is to show that the group of orientation-preserving isometries of , that is
is a simple group. Here we use a geometric point of view, mixing the proofs that can be found in Stillwell’s book, Naive Lie theory, and Perrin’s book, Cours d’algèbre. We begin with an easy structure lemma:
Lemma 1: If , there exists an orthonormal basis in which may be written as
for some .
Proof. First, notice that
hence ie. there exists such that . Let . Then for all ,
so is stable under . Therefore, may be written as in an orthonormal basis; in particular, because and , .
If , the equality becomes . The two first equations allow us to set for some . Then, the two last equations give
, that is .
Therefore, and may be written as the mentionned matrix.
Definition: If is as in the lemma, we say that is a rotation of axis and angle .
Lemma 2: Let and . Suppose that is a rotation of axis . Then is a rotation of axis .
Proof. Let . Then and is stable under . Therefore, according to lemma 1, is a rotation of axis .
Lemma 3: Following the notations given on the figure below, the rotation around of angle composed with the rotation around of angle is a rotation around whose angle equals to twice the angle at of the spherical triangle .
Proof. Let be two planes intersecting along a line . Let be the reflections with respect to respectively. Then fixes and it is not difficult to see that the restriction of to the plane normal to is a rotation of angle twice the angle between and . Therefore, according to lemma 1, is a rotation of axis and of angle .
Thus, if denotes the rotations around of angles respectively, and if denotes the reflections with respect to respectively, we can write
hence , that is is a rotation around of angle twice the angle at of the spherical triangle .
Now, we give two lemmas to characterize the natural action of on the sphere .
Lemma 4: The action is transitive, ie. for every there exists such that .
In fact, it is a special case of the following lemma, which states that the action is “as -transitive as possible”.
Lemma 5: The action is isometrically 2-transitive, ie. for every satisfying there exists such that and .
Proof. Let be a rotation sending to , and be a rotation of axis sending to . Then and
Theorem: is a simple group.
Proof. Let be a non-trivial normal subgroup. We first want to show that the action is transitive. Let be a rotation of axis for some and let . Because , , so .
Notice that sends the meridian passing through to the meridian passing through , so runs over when runs over : tends to (resp. ) when tends to the north pole (resp. to the equator). More precisely, if , it is possible to find so such that
Because is linear and , the above equation is equivalent to
, hence .
Let such that . We just saw that there exists such that . According to lemma 5, there exists such that and . Then because is a normal subgroup and .
Therefore, we proved that the action is transitive “for the small distances”. Now, if are any points, clearly there exists a sequence of points satisfying for all . In particular, for all , there exists such that , and finally is an element of sending to . Thus, the action is transitive.
In particular, there exist and such that ; such an element of , with as an eigenvalue, is called a codimension-two reflection. Clearly, using lemmas 1 and 2, two codimension-two reflections are always conjugated. Therefore, since is a normal subgroup, any codimension-two reflection belongs to . We deduce, according to lemma 3 and using the notations of the figure below, that the rotation of angle around belongs to .
Clearly, when runs over the equator, runs over . Therefore, contains rotations of any angle. Because two rotations of the same angle are conjugated and is a normal subgroup, we deduce that any rotation belongs to , that is .
Nota Bene: The conclusion of our proof, based on lemma 3, may be replaced with an algebraic argument, stating that the codimension-two reflections generate . Therefore, in order to show that a normal subgroup is , it is sufficient to find only one codimension-two reflection belonging to it.
Another classical proof is to find a codimension-two reflection in a nontrivial normal subgroup by introducing the map
for some fixed . By connectedness, for some , because for all there exists such that (lemma 1). If , then for all , that is is in the center of , which is trivial. Therfore, and there exist and such that
Then is a rotation of angle and is a codimension-two reflection.
Finally, a proof based on the isomorphism exhibit in Fundamental group of SO(3) and Quaternions can be found in Artin’s book, Algebra.