Abelian Groups

A group is considered abelian if it is also commutative. This means that for all a, b \in G, a*b = b*a.

For example \mathbb{Z}_4 is an abelian groups since modular arithmetic is also commutative. How set \{1,-1,i,-i\} under multiplication also forms an abelian group since multiplication is commutative.

Commutativity can easily be taken for granted since addition and multiplication for numbers and complex numbers, even modular arithmetic operations are commutative. There are some operations such as matrix multiplication, are not commutative.

The rotation SO(3) group, a three dimensional rotation group, is not an abelian group since matrix multiplication is not commutative. However not all matrix groups are non abelian, some such as the SO(2) rotation group are abelian.

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  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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