Question

Example of abelian group that is not cyclic

Answer 1

You simply need an abelian group of order 1212, with no elements of order 1212. G=Z6×Z2G=Z6×Z2 will do (where ZnZndenotes the cyclic group of order nn). As a direct product of cyclic (so abelian) groups, GG is again abelian. Given any element (x,y)∈G(x,y)∈G, the order of (x,y)(x,y) will be the least common multiple of the orders of x,y.x,y. The order of xx must divide 66and the order of yy must divide 2,2, so the order of (x,y)(x,y) is at most lcm(6,2)=6.lcm⁡(6,2)=6. But |G|=|Z6|⋅|Z2|=6⋅2=12.|G|=|Z6|⋅|Z2|=6⋅2=12.Since no element of GG has order 12,12,then GG is not cyclic.