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Homework Statement
Determine how many non-isomorphic (and which) abelian groups there are of order 54.
Determine which of these groups the factor group Z6 x Z18 / <(3,0)> is isomorphic to.
Homework Equations
The Attempt at a Solution
Fundamental theorem for abelian groups gives:
54 = 2*3^3 and then the groups are
Z2 x Z3 x Z3 x Z3
Z2 x Z9 x Z3
Z2 x Z27
For the second part: <(3,0)> = { (0,0), (3,0) } so we get that the order of our group is (6*18)/2 = 54 as expected. But how do I decide which group it is isomorphic to? I have tried looking at cosets like (1,0) + <(3,0)> but don't seem to get anywhere...