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have expanded the statement
The n-spheres are coset spaces of orthogonal groups
S^n \;\simeq\; O(n+1)/O(n),.$
to :
Similarly for the corresponding special orthogonal groups
$S^n \;\simeq\; SO(n+1)/SO(n)$and spin groups
$S^n \;\simeq\; Spin(n+1)/Spin(n)$and pin groups
$S^n \;\simeq\; Pin(n+1)/Pin(n) \,.$added some more words to the section Coset space structure
added this reference in the section about group actions on spheres:
added pointer to
which according to p.2 of
is the actual origin of the classification of the coset space realizations of $n$-spheres, which I have added now as a Prop:
The connected Lie groups with effective transitive actions on n-spheres are precisely (up to isomorphism) the following:
with
$\begin{aligned} SO(n)/SO(n-1) & \simeq S^{n-1} \\ U(n)/U(n-1) & \simeq S^{2n-1} \\ SU(n)/SU(n-1) & \simeq S^{2n-1} \\ Sp(n)/Sp(n-1) & \simeq S^{4n-1} \\ Sp(n)\cdot SO(2)/Sp(n-1)\cdot SO(2) & \simeq S^{4n-1} \\ Sp(n)\cdot Sp(1)/Sp(n-1)\cdot Sp(1) & \simeq S^{4n-1} \\ G_2/SU(3) & \simeq S^6 \\ Spin(7)/G_2 & \simeq S^7 \\ Spin(9)/Spin(7) & \simeq S^{15} \end{aligned}$1 to 5 of 5