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Algebra/Chapter 27/Groups

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------------------------ Algebra
Chapter 27: Group Theory
Section 3: Groups
Lagrange's Theorem

27.3: Groups


Definition of a Group

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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 such that

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