Let and be positive integers and . Let and . Let be the central product of and with an element of order identified; this group has the presentation . Describe as a direct product of two cyclic groups.
[Proof suggested by Fred H]
Note that and are relatively prime. By Bezout’s identity, there exist such that . Let .
Now . (For the second equality,we used the fact that and commute.) Similarly, . In particular, generates . By Lagrange’s Theorem, ; so .
We can write as a direct product of cyclic groups as either or . (Since, for instance, and are relatively prime.)