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> In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play.

Number theory is another example, finding roots of polynomials was already mentioned.

> Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).

It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)