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Topology studies the invariants of a system during a deformation, and Group Theory can be used to study the symmetry, which can also be considered as some sort of invariant. Is there any relationship between them?

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    Yes, definitely. At least, there's a symbiotic relationship between the two. See the tag [tag:geometric-group-theory], for instance. – 2017-02-17
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    You can also look at topological groups:http://math.stackexchange.com/questions/tagged/topological-groups – 2017-02-17
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    [Topological methods in group theory](http://www.springer.com/us/book/9780387746111). Also: before asking at MSE you should at least try to google your question, say , will give you quite a bit of material to ponder. – 2017-02-17

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An important relationship occurs for Lie groups, which are both groups and differentiable manifolds. A simple example is $GL(n,\mathbb{R})$, the group of invertible $n \times n$ matrices

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    This is a special case of a "topological group", a subject on which whole books have been written. These are groups that have a topology which makes all group operattions continuous. They have a lot of extra structure. – 2017-02-17