I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises are really similar to the ones presented here.
1) Are $\mathbb{Z}_{10}$ and $\mathbb{Z}_2 \times \mathbb{Z}_5$ isomorphic?
2) Are $\mathbb{Z}_{54}$ and $\mathbb{Z}_6 \times \mathbb{Z}_9$ isomorphic?
On
For your first question, consider the element $(1,1)$ in $\Bbb Z_2 \times \Bbb Z_5$ (under addition modulo $2$ in the first coordinate, and modulo $5$ in the second coordinate). What is its order?
The second question is more subtle-consider what happens to the element $(a,b) \in \Bbb Z_6 \times \Bbb Z_9$, when we add it to itself $18$ times (note that $6|18$ and $9|18$). What does this tell you about the maximum possible order of elements of $\Bbb Z_6 \times \Bbb Z_9$?
1) Since $2$ and $5$ are coprime and $2 \times 5 = 10$, this is a direct result of the Chinese remainder theorem : https://en.wikipedia.org/wiki/Chinese_remainder_theorem#Theorem_statement
You can do the same for the second case. ($6$ and $9$ not coprime so ...)
See page 12 for a proof : http://www.math.columbia.edu/~rf/numbertheory.pdf