One of the most fundamental algebraic structures in mathematics is the group. A group is a set of elements paired with an operation that satisfies the following four conditions:
I. It is closed under an operation (represented here by “+”, although it does not necessarily mean addition): For all elements a and b in the set S, a+b is also in S.
II. It contains an identity element (often written as “e”): There is some element e in the set S such that for every element a in S, a+e = e+a = a.
III. The operation is associative: For all a, b, and c in the set S, (a+b)+c = a+(b+c).
IV. Inverses exist: For every element a in the set S, there is an a-1 in S such that a+a-1 = a-1+a = e
II. It contains an identity element (often written as “e”): There is some element e in the set S such that for every element a in S, a+e = e+a = a.
III. The operation is associative: For all a, b, and c in the set S, (a+b)+c = a+(b+c).
IV. Inverses exist: For every element a in the set S, there is an a-1 in S such that a+a-1 = a-1+a = e
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