I found this problem: find the types of isomorphism between abelian groups of order 144. Up until now I only found that G, an abelian group of 144 elements, can only be isomorphic with either Z2 x Z2 x Z2 x Z2 x Z3 x Z3, Z2 x Z8 x Z3 x Z3, Z4 x Z4 x Z3 x Z3, Z2 x Z2 x Z4 x Z3 x Z3 or Z16 x Z3 x Z3, and again all of those but with Z9 instead of Z3 x Z3, but I don't know anything about the morphisms between them.
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How many elements of order $2$, $4$, $8$ and $16$ do you have in each of them? Why does this matter for the isomorphism question? – Clément Guérin May 18 '18 at 12:37
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5I think the exercise is more to find the isomorphism types of Abelian groups of order 144, that is, find the possible Abelian groups of order 144 up to isomorphism. More specifically, you should give an argument as to why these are candidates for such groups, and then argue that they are pairwise non-isomorphic. – Edward Evans May 18 '18 at 12:37
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Here's a MathJax tutorial :) – Shaun May 18 '18 at 15:16
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The possible abelian groups of order $144=2^4 \cdot 3^2$ correspond to the combinations of additive partitions of the exponents $4$ and $2$. There are thus $5 \cdot 2=10$ possible abelian groups of order $144$. Just complete the table below. No two entries are isomorphic. $$ \matrix{ & 2 & 1+1 \\ 4 & C_{16} \times C_9 & C_{16} \times C_3 \times C_3 \\ 3 + 1 & C_{8} \times C_ 2 \times C_9 & C_{16} \times C_3 \times C_3 \\ 2 + 2 \\ 2 + 1 + 1 \\ 1 + 1 + 1 + 1 & & C_ 2 \times C_ 2 \times C_ 2 \times C_ 2 \times C_ 3 \times C_ 3\\ } $$ Note that you can combine $2$-factors with $3$-factors, but won't get anything new. For instance $$ C_{8} \times C_{18} \cong C_{8} \times C_2 \times C_9 \cong C_{2} \times C_{72} $$
lhf
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