cyclic group

plural cyclic groups

1. group theoryA group generated by a single element.

   1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22, Proposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group.

   2 more examples

   If u#58;G#95;1#92;rightarrow#92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125; and v#58;G#95;2#92;rightarrow#92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125; are isomorphisms of two cyclic groups with #92;mathbb#123;Z#125;#47;m#92;mathbb#123;Z#125;, then v#123;-1#125;#92;circu#58;G#95;1#92;rightarrowG#95;2 is an isomorphism.

   2002, Serge Lang, Algebra, 3rd edition, Springer, page 24

   2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152, Cyclic groups are the simplest groups imaginable.

   Synonyms: monogenous group