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

Nann 2020-10-25 Answered
Is Z3  Z9, isomorpbic to
Decide and answer why exactly?
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Answered 2020-10-26 Author has 75 answers

To decide if the first group (the direct sum) is isomorpbic to Z27,
Description of the groups whose isomorphism is under question. The question arises because both (the direct sum as well as Z27 contain the same number of elements:27). We will show that these two groups are NOT isomorphic
Now, Z3  Z={(x, y) : x  Z3, y  Z9}
Where, Z3={0, 1, 2}(addition modulo 3)
Z9={0, 1, 2, 3,  8}(addition modulo 9) whereas,
Z27={0, 1, 2, 3,  26}(addition modulo 27)
First observe that the direct sum is not a cyclic group.
Now, Z3  Z9={(x, y) : x  Z3, y  Z9}
Now, 9(x, y)=(9x, 9y)=(0, 0),A(x, y) Z3  Z9
So any element has order at most 9
So, Z3  Z9 is not cycllc.
(otherwise, E(x, y)  Z3 oi Z9 with order 27)
On the other hand Z27 is a cyclic group, for example , with 1 as a generator. So the given two groups are not isomorphic.

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