the difference between a ring and a group, and examples of both (other than numbers)
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One intuitive way to think about groups is that they are sets of symmetries of things. For example, if I have a square, I can rotate it by 90 degrees and it will look the same as if I hadn’t rotated it at all – so rotation by 90 degrees is an element of the group of symmetries of a square, among other things.
The most important example of a group is the set of permutations of n things- all possible reorderings of {1,2,3,…,n}. It contains n! elements and is called the symmetric group, and it’s a theorem that every group of size n is contained within the symmetric group of size n.
Importantly, groups only have one “operation” (way of combining elements), usually called multiplication (but sometimes addition).
A ring is a group but with some additional structure(or alternately, a field with fewer requirements). Namely, a ring is an abelian group under addition (ie, a+b=b+a for all a, b in the ring). It also has a multiplication which has to be associative and distribute over addition. Unlike a field, the multiplication need not commute, and you don’t require multiplicative inverses. Examples include matrices under matrix addition and matrix multiplication, and polynomials, among many others.
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