I am given that G is a finite abelian group that contains
8 elements of order three
18 elements of order nine
and an identity
And am told to describe of possibilities for G, by giving explicit decomposition into cyclic groups up to an isomorphism
So it seems like there are three possibilities for G
(is this isomorphic to ?)
Now G cannot be , for G has no element of order 27.
I also think G cannot be , since no element of that group has order nine. I pretty sure every element of this group must have order less than or equal to 3, since if , then
So I am left with . Is there a quicker way to test the elements of this group besides considering the order of all 27 elements?
i am pretty sure the element (1,a) \in , would have order 9, but I dont think there's 18 of those.
Thank you for your time.