Cyclic Groups

Another type of group is called the cyclic group, which is a group generated by a single element. For example \mathbb {Z}_8 is a cyclic group because all of the elements in \mathbb{Z}_8 can be generated by 1 by modular arithmetic. \mathbb{Z}_8 = <1> = \{1, 1+1, 1+1+1 ... \}.

More formally let G be a group and g \in G, G is a cyclic group if G = <g> = \{g^n | n \in N\}; g^n does not mean exponentiation in that g^2 does not mean g multiplied by g, it means g*g where * can be addition or any defined binary operation.  For example g^5 = g*g*g*g*g.

Another example is the group H = \{1,-1,i,-i\} = <i> = \{i, i^2, i^3, i^4\} = \{i, -1, -i, 1\}.

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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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