1 Prove that a factor group of a cyclic group is cyclic Use

1. Prove that a factor group of a cyclic group is cyclic. (Use the definition of cyclic group, factor group)

Solution

solution

Suppose that G = hai and that H G. An element of G/H has the form gH for some g H. Each element g can be written as a k for some k. Now

a^k(H) = (aH)^k ,

(as can be seen by an easy inductive proof, and the definition of the product in G/H.)

Therefore G/H = haHi is cyclic, as required