To decide if the first group (the direct sum) is isomorpbic to
Description of the groups whose isomorphism is under question. The question arises because both (the direct sum as well as contain the same number of elements:27). We will show that these two groups are NOT isomorphic
Where, (addition modulo 3)
(addition modulo 9) whereas,
(addition modulo 27)
First observe that the direct sum is not a cyclic group.
So any element has order at most 9
So, is not cycllc.
(otherwise, with order 27)
On the other hand is a cyclic group, for example , with 1 as a generator. So the given two groups are not isomorphic.