Question

Find the elementary divisors of the given group:${\mathrm{\mathbb{Z}}}_6\oplus {\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{18}$

Short answer

The elementary divisors of the group${\mathrm{\mathbb{Z}}}_6\oplus {\mathrm{\mathbb{Z}}}_{12}\oplus {\mathrm{\mathbb{Z}}}_{18}$ is $\{2,2,4,3,3,9\}$.

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 ℤ6⊕ℤ12⊕ℤ18

The prime factors of 6 is $6=2\times 3$12 is $12=2^2\times 3$ , 18 is $18=2\times 3^2$ .

Hence, the group can be written as

${\mathbb{Z}}_6\oplus {\mathbb{Z}}_{12}\oplus {\mathbb{Z}}_{18}\cong ({\mathbb{Z}}_2\oplus {\mathbb{Z}}_3)\oplus ({\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_3)\oplus ({\mathbb{Z}}_2\oplus {\mathbb{Z}}_{3^2})$

Therefore, the elementary divisors of the group ${\mathbb{Z}}_6\oplus {\mathbb{Z}}_{12}\oplus {\mathbb{Z}}_{18}$isrole="math" localid="1652853786736" $\{2,2,4,3,3,9\}$ .