The exercise asks to show that the given identification is an isomorphism of LCA groups (so the identification need to be group isomorphisms and also topological homeomorphisms).

]]>*[Corrected, thanks – T.]*

The link is now not working due to the close of google plus. Do you still have it somewhere?

*[ Now at https://terrytao.wordpress.com/advice-on-writing-papers/on-compilation-errors-in-mathematical-reading-and-how-to-resolve-them/ -T.]*

Well, it depends on what you mean by “real” or “interesting”. Technically, any homogeneous space of cardinality less than or equal to the continuum can be embedded by some set-theoretic function into or , and any homogeneous space that is a smooth manifold can be embedded smoothly into a sufficiently high dimensional Euclidean space (Whitney embedding theorem). But in addition to the circle and sphere, other quadric surfaces can be viewed as quotients. For instance the unit hyperboloid can be viewed as a quotient . An ellipsoid associated to some positive definite form can be viewed as a quotient , where is the group of three-dimensional unimodular linear transformations preserving , and is the stabiliser of some point on the ellipse in . An elliptic curve in Weierstrass form is almost a group inside (though one has to add the point at infinity), though a real elliptic curve can also be identified with either one or two circles which can also be embedded into if desired. Then of course there are the finite examples; for instance, the vertices of a regular polyhedron can be viewed as a homogeneous space formed by quotienting the symmetry group of the polyhedron by the stabiliser of a vertex.

]]>A quotient of two groups only has a natural group structure in the case when is a normal subgroup of . Otherwise it is simply a set (though it can inherit a topology or a sigma algebra from by the usual quotient construction, and under reasonable compactness-type conditions one can also inherit a Haar measure).

]]>To put it in another way, if one proves a result for any homogeneous space such that is compact Hausdorff and is closed topological groups, how many interesting “real” examples of the space are there on and besides the unit circle and the sphere?

]]>I may misunderstand something: one can write ; why “the sphere is not a group”?

]]>Of course one can write . How about ellipses? Or a square? A rectangle?

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