A Crash Course in Groups

Groups provide structure because of this groups are used to represent phenomena in Physics, Chemistry and Computer Science. This tutorial is focused on giving the basic definitions of different types of groups.

This tutorial will build up on ideas from A Crash Course in Set Notation and A Crash Course in Modular Arithmetic.

What is a Group?

A group is a set with binary operation. Groups are generally denoted as <G,*> where G is a set and * is the binary operation. The binary operation does not need to be multiplication or addition, it needs to be any well defined operation that takes two elements from set G and produces one element in set. In order to be considered a group, the set G and binary operation * must following a list of rules.

  1. Closure : For all a, b \in G, a*b \in G.
  2. Identity : There exists element e \in G such that for all g \in G e*g = g (For addition the identity is 0, for multiplication the identity is 1)
  3. Associativity : For all a, b, c \in G, (a*b)*c = a*(b*c)
  4. Inverse : For all g \in G there exists an element g^{-1} \in G such that g*g^{-1} = e.


  1. What is a Group
  2. Example of Groups
  3. Abelian Groups
  4. Subgroups and Cosets
  5. Cyclic Groups
  6. Applications

Filed Under: Crashes CoursesMathematics

Sources Used

  • Abstract Algebra, Theory and Application by Thomas W. Judson
  • Abstract Algebra by David S. Dummit and Richard M. Foote
  • Algorithms by Sanjoy Dasgupta, Christos Padadimitrou and Umesh Vazirani

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