Question

List all abelian groups (up to isomorphism) of the given order:72

Short answer

The abelian groups of order 72 are

$\left\{\begin{array}{l}{\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2},\\ {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\end{array}\right\}$.

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 72

The number 72 can be written as a product of prime powers in six different ways: they are $\{(2^3\cdot 3^2),(2\cdot 2^2\cdot 3^2),(2\cdot 2\cdot 2\cdot 3^2),(2^3\cdot 3\cdot 3),(2\cdot 2^2\cdot 3\cdot 3),(2\cdot 2\cdot 2\cdot 3\cdot 3)\}$

Therefore, the abelian group can be written as follows:

$\begin{array}{l}{\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\end{array}$

Hence, the abelian groups of order 72 are

$\left\{\begin{array}{l}{\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2},\\ {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\end{array}\right\}$