Question

Find the elementary divisors of the given group:${\mathit{\mathbb{Z}}}_{\mathbf{10}}\mathbf{\oplus }{\mathit{\mathbb{Z}}}_{\mathbf{20}}\mathbf{\oplus }{\mathit{\mathbb{Z}}}_{\mathbf{30}}\mathbf{\oplus }{\mathit{\mathbb{Z}}}_{\mathbf{40}}$

Short answer

The elementary divisors of the group ${\mathbb{Z}}_{10}\oplus {\mathbb{Z}}_{20}\oplus {\mathbb{Z}}_{30}\oplus {\mathbb{Z}}_{40}$is$\{2,5,4,5,8,5,2,3,5\}$.

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 ℤ10⊕ℤ20⊕ℤ30⊕ℤ40

The prime factors of 10 is $10=2\times 5$20 is $20=2^2\times 5$ , 30 is $30=2\times 3\times 5$ , 40 is $40=2^3\times 5$.

Hence, the group can be written as:

${\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}\cong ({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_5)\oplus ({\mathrm{\mathbb{Z}}}_{2^2}\times {\mathrm{\mathbb{Z}}}_5)\oplus ({\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_3\times {\mathrm{\mathbb{Z}}}_5)\oplus ({\mathrm{\mathbb{Z}}}_{2^3}\times {\mathrm{\mathbb{Z}}}_5)$

Therefore, the elementary divisors of the group ${\mathrm{\mathbb{Z}}}_{10}\oplus {\mathrm{\mathbb{Z}}}_{20}\oplus {\mathrm{\mathbb{Z}}}_{30}\oplus {\mathrm{\mathbb{Z}}}_{40}$is:

$\{2,5,4,5,8,5,2,3,5\}$.