Question

Find the elementary divisors of the given group:${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}$

Short answer

The elementary divisors of the group ${\mathbb{Z}}_{12}\oplus {\mathbb{Z}}_{30}\oplus {\mathbb{Z}}_{100}\oplus {\mathbb{Z}}_{240}$ is $\{4,3,16,3,5,2,3,5,4,25\}$.

Step-by-step solution

Step-by-Step Solution Step 1: Finite Abelian Group

Every finite abelian group G is the direct sum of cyclic groups, each of prime power orders.

Elementary divisors of ℤ12⊕ℤ30⊕ℤ100⊕ℤ240

The prime factors of 12 is $12=2^2\times 3$ , 30 is $30=2\times 3\times 5$, 100 is $100=2^2\times 5^2$, 240 is $240=2^4\times 3\times 5$.

Hence, the group can be written as:

${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}\cong ({\mathrm{\mathbb{Z}}}_{2^2}\times {\mathrm{\mathbb{Z}}}_3)\oplus ({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_5)\oplus ({\mathrm{\mathbb{Z}}}_{2^2}\times {\mathrm{\mathbb{Z}}}_{5^2})\oplus ({\mathrm{\mathbb{Z}}}_{2^4}\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_5)$

Therefore, the elementary divisors of the group${\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{100}\oplus {\mathrm{\mathbb{Z}}}_{240}$ is$\{4,3,16,3,5,2,3,5,4,25\}$ .