Group

Pronunciation: /grup/ ?

A group is a set with an operation defined on members of that set. The operation must meet the requirements of closure, associativity, identity and invertibility.

Example: the set of real numbers under addition is a group since:

  1. Closure: for any real numbers a and b, a + b = c if and only if c is a real number.
  2. Associativity: for any real numbers a, b and c, a+(b+c) = (a+b)+c.
  3. Identity: 0 is the additive identity for real numbers because for any real number a, a + 0 = a and 0 + a = a.
  4. Invertibility: for every real number a, there is an additive inverse -a such that a + (-a) = 0.

A commutative group is a group where the operation is also commutative. If, for any members of the group S, a and b, a*b = b*a, then group S is a commutative group. Commutative groups are also called Abelian groups.

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Group. 2010-06-12. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/g/group.html.

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2010-06-12: Initial version (McAdams, David.)

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