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Let us have a geometry with a set X and a group G. What if X itself can be endowed with a group structure to be isomorphic to G? Can we gain something from this?

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    Can you please define "a geometry with a set $X$ and a group $G$"? After that, can you please improve the title by making it more informative and understandable?2011-09-11

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This is in fact very well-studied. Examples are:

A topological group is both a topological space and an abstract group, and the two structures are compatible.

A Lie group is both a smooth manifold and an abstract group, and the two structures are compatible.

A algebraic group is (to be simple) both an algebraic variety(an idea from algebraic geometry) and also an abstract group, and the two structures are compatible.

Etc, etc...

You can find the detailed definitions from wikipedia, if you want.