This problem can be solved by looking at corollary 2 of theorem 8.2. According to this corollary Z_a x Z_b is isomorphic to Z_ab only if a and b are relatively prime, however since gcd(3,9) = 3 this is not the case. Therefore, Z_3 x Z_9 is not isomorphic to Z_27.
--Jniederh 22:44, 23 February 2009 (UTC)
Also, Z_27 has an element of order 27, but the highest order of an element in Z_3 x Z_9 is 9. --sgrosenb
