I wish to find the invariant factors for the finite, abelian group:
$B = \Bbb{Z}_6 \oplus \Bbb{Z}_{100} \oplus \Bbb{Z}_{45}$
of the order $2^3 \cdot 3^3 \cdot 5^3 = 27000$.
I have prior to this only been used to finding invariant factors for standard abelian groups of the form i.e. $A = \Bbb{Z}_{p^2 q} \oplus \Bbb{Z}_{p} \oplus \Bbb{Z}_{qr}$ of the order $p^3 q^2 r$.
I know that for a group of order $x$, the elementary divisors are all combinations of the prime factors and the invariant factors are the combinations of these prime factors, when all co-prime numbers are multiplied from elementary divisors, but I'm not sure how to attack this problem, with this information.
Can anybody drop a hint?
Best Answer
First use the Chinese Remainder Theorem to split each factor in $B$ into $p$-groups: $$ B = \Bbb{Z}_6 \oplus \Bbb{Z}_{100} \oplus \Bbb{Z}_{45} \cong \Bbb{Z}_2 \oplus \Bbb{Z}_3 \oplus \Bbb{Z}_{4} \oplus \Bbb{Z}_{25} \oplus \Bbb{Z}_{9} \oplus \Bbb{Z}_{5} $$
Now list the powers of each prime in increasing order right justified and multiply them vertically (again using the Chinese Remainder Theorem): \begin{array}{r} 2 & 4 \\ 3 & 9 \\ 5 & 25 \\ \hline 30 & 900 \\ \end{array} This gives the invariant factors $\Bbb{Z}_{30} \oplus \Bbb{Z}_{900}$.