Question

List all abelian groups (up to isomorphism) of the given order:12

Short answer

The abelian groups of order 12 are${\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3$,${\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3$.

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 12

The number 12 can be written as a product of prime powers in two different ways: $12=2^2\cdot 3$ and$12=2\cdot 2\cdot 3$.

Hence, by the above theorem, the abelian groups can be written as ${\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3$, ${\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3$.

Therefore, the abelian groups of order 12 are ${\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3$,${\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3$.