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Mathematical structures such as Numbers, Matrices, Vectors are all groups, rings, or similar?

Can you provide more examples of more thease.

Do operations such as addition occur (although in different ways) between any combination of two of thease, for example a number and a vector, or only between two of the same type, for example two vectors?

Is there a turm for the operations that can occur between groups? I think they are + and ×.

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    Have you tried the Wikipedia article on group theory?2012-07-15
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    Yes. I didn't have much luck with it. I'm looking for a simpler explanation, and I don't need to know how to use them mathematically, a high level description will do.2012-07-15

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