Pronunciation: /grup/ Explain

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.


  1. McAdams, David E.. All Math Words Dictionary, group. 2nd Classroom edition 20150108-4799968. pg 88. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Group. 4/21/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

4/21/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/10/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)

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