Question

Find the elementary divisors of the given group:${\mathrm{\mathbb{Z}}}_{250}$

Short answer

The elementary divisors of the group${\mathrm{\mathbb{Z}}}_{250}$is$\{2,125\}$.

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 ℤ250

The prime factors of 250 are $250=2\times 5^3$

Hence, the group can be written as ${\mathrm{\mathbb{Z}}}_{250}\cong {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{5^3}$

Therefore, the elementary divisors of the group ${\mathrm{\mathbb{Z}}}_{250}$is$\{2,125\}$.