How Can Analogies Simplify Group, Ring, and Field Concepts?

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As we know for the average Undergrad attempting to grasp (and understand) these abstract mathematical concepts can be challenging to say the least. I was (and still am in some sense :P) in that boat. Does anyone have any Analogies or creative ways of explaining these and getting their meaning across while retaining some type of concrete idea?
 
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The best way would be to start with simple examples of groups/fields/rings.
 

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