Question

List all abelian groups (up to isomorphism) of the given order:15

Short answer

The abelian group of order 15 is${\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5$.

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.

Abelian groups of order 15

The number 15 can be written as a product of prime powers as,$15=3\cdot 5$.

Hence, by the above theorem, the abelian groups can be written as: ${\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5$.

Therefore, the abelian group of order 15 is${\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5$ .