Group (mathematics)

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This article is about the basic notions. For advanced topics, see Group theory.
A Rubik's Revenge cube. All the possible positions of the cube, if paired with a suitable operation, would form a group.

In mathematics, a group is a kind of algebraic structure. A group has a set and a operation. The group's operation can combine any two elements of the group's set. This forms a third element.

Instead of "an element of the group's set", mathematicians usually save words. They say "an element of the group".

Mathematicians use capital letters to stand for groups. They often use G, H, or K.

They use lower-case letters to stand for group elements. To save words, they say "a is in G" to mean "a is an element of G".

They write group operations with symbols like • or *, or by writing two elements next to each other. So "ab", "a * b", and "ab" can all mean "the element formed when the group's operation combines a and b".

Not every set and operation make a group. A group's set and operation must obey some special rules. These are called group axioms. This list has each axiom twice, once in words, and once in mathematical symbols.

  • Closure: When a group's operation combines two elements, the element that is formed must also be part of the group.
    • For all a, b in G, ab is also in G.
  • Identity element: One element of the group is special. It is called the identity element. If the operation combines the identity element and any second element, the answer will be that second element.
    • There exists e in G, so that for all a in G, ea = ae = a.
  • Associativity: When the operation is used twice to combine three elements, it does not matter in what order they are done, because the answer will be the same.
    • For all a, b and c in G, (ab) • c = a • (bc).
  • Inverse element: Every element in the group has another element in the group that is called its inverse. When the operation combines any element and its inverse, the answer is the identity element.
    • For each a in G, there must exist b in G, so that ab = ba = e, where e is G's identity element.

In a group, it is not always true that ab = ba for every a and b in the group. In other words, a group's operation does not have to be commutative.

If a group's operation is commutative, we call it an abelian group.

The number of elements in a group is called the group's order.

One everyday example of a group is the set of integers and the addition operation. It is easy to see that when you add two integers, the result is always an integer. So closure is true. The identity element for this group is zero. Because the order of additions does not matter (or in other words, a + (b + c) = (a + b) + c), associativity is true. And the inverse of any integer is its negative value. This group is also an abelian group. It has infinite order, because the integers are a countably infinite set.