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

Is isomorpbic to
${Z}_{27}?$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

likvau

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,
First observe that the direct sum is not a cyclic group.
Now,
Now,
So any element has order at most 9
So, is not cycllc.
(otherwise, 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.