Question

List all abelian groups (up to isomorphism) of the given order:90

Short answer

The abelian groups of order 90 are$\{{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2}\oplus {\mathrm{\mathbb{Z}}}_5,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\}$.

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 90

The number 90 can be written as a product of prime powers in two different ways which are $\{(2\cdot 3^2\cdot 5),(2\cdot 3\cdot 3\cdot 5)\}$.

Therefore, the abelian groups of order 90 are$\{{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2}\oplus {\mathrm{\mathbb{Z}}}_5,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_5\}$.