Algebra/Chapter 27/Groups

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Algebra Chapter 27: Group Theory Section 3: Groups

Lagrange's Theorem

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Definition of a Group

In standard terms, a group G is a set equipped with a binary operation • such that the following properties hold:

1. The binary operation is closed. That means, for any two values a and b in G, the combined value a • b is also in G.
2. The binary operation is associative. For any values a, b, c in G, a • (b • c) = (a • b) • c.
3. There exists a unique identity element e in G such that for all values a in G, a • e = a = e • a.
4. There exists a unique inverse element $a^{-1}$ such that $a^{-1}\bullet a=e$

If the binary operation is commutative, or b • a = a • b, then the group is said to be Abelian.