Is Z_{3} oplus Z_{9}, isomorpbic to Z_{27}? Decide and answer why exactly?

To decide if the first group (the direct sum) is isomorpbic to ${\mathbb{Z}}_{27},$
Description of the groups whose isomorphism is under question. The question arises because both (the direct sum as well as ${\mathbb{Z}}_{27}$ contain the same number of elements:27). We will show that these two groups are NOT isomorphic
Now, ${\mathbb{Z}}_3\oplus \mathbb{Z}=\{(x,y):x\in {\mathbb{Z}}_3,y\in {\mathbb{Z}}_9\}$
Where, ${\mathbb{Z}}_3=\{0,1,2\}$(addition modulo 3)
${\mathbb{Z}}_9=\{0,1,2,3,\cdots 8\}$(addition modulo 9) whereas,
${\mathbb{Z}}_{27}=\{0,1,2,3,\cdots 26\}$(addition modulo 27)
First observe that the direct sum is not a cyclic group.
Now, ${\mathbb{Z}}_3\oplus {\mathbb{Z}}_9=\{(x,y):x\in {\mathbb{Z}}_3,y\in {\mathbb{Z}}_9\}$
Now, $9(x,y)=(9x,9y)=(0,0),\mathbb{A}(x,y)\in {\mathbb{Z}}_3\oplus {\mathbb{Z}}_9$
So any element has order at most 9
So, ${\mathbb{Z}}_3\oplus {\mathbb{Z}}_9$ is not cycllc.
(otherwise, $\mathbb{E}(x,y)\in {\mathbb{Z}}_{3}oi{\mathbb{Z}}_9$ with order 27)
On the other hand ${\mathbb{Z}}_{27}$ is a cyclic group, for example , with 1 as a generator. So the given two groups are not isomorphic.