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:<math>[cq_0,cq_1\ldots,cq_n]</math>.
In the language of [[Group action (mathematics)|group action]]s, <math>\mathbb{HP}^n</math> is the [[orbit space]] of <math>\H^{n+1}\setminus\{(0,\ldots,0)\}</math> by the action of <math>\H^{\times}</math>, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside <math>\H^{n+1}</math> one may also regard <math>\mathbb{HP}^{n}</math> as the orbit space of <math>S^{4n+3}</math> by the action of <math>\text{Sp}(1)</math>, the group of unit quaternions.<ref>Gregory L. Naber, ''Topology, geometry, and gauge fields: foundations'' (1997), p. 50.</ref> The sphere <math>S^{4n+3}</math> then becomes a [[principal bundle|principal Sp(1)-bundle]] over <math>\mathbb{HP}^n</math>:
:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb{HP}^n.</math>
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