Question

List all abelian groups (up to isomorphism) of the given order:144

Short answer

The abelian groups of order 144 are:

$\left\{\begin{array}{l}{\mathbb{Z}}_{2^4}\oplus {\mathbb{Z}}_{3^2},{\mathbb{Z}}_2\oplus {\mathbb{Z}}_{2^3}\oplus {\mathbb{Z}}_{3^2},{\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_{3^2},{\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_{3^2},\\ {\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_{3^2},{\mathbb{Z}}_{2^4}\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,{\mathbb{Z}}_2\oplus {\mathbb{Z}}_{2^3}\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,{\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,\\ {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,{\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_{2^2}\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\end{array}\right\}$

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 144

The number 144 can be written as a product of prime powers in 10 different ways:

$\left\{\begin{array}{l}(2^4\cdot 3^2),(2\cdot 2^3\cdot 3^2),(2\cdot 2\cdot 2^2\cdot 3^2),(2\cdot 2\cdot 2\cdot 2\cdot 3^2),(2^2\cdot 2^2\cdot 3^2)\\ (2^4\cdot 3\cdot 3),(2\cdot 2^3\cdot 3\cdot 3),(2\cdot 2\cdot 2^2\cdot 3\cdot 3),(2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3),(2^2\cdot 2^2\cdot 3\cdot 3)\end{array}\right\}$

Therefore, the abelian group can be written as follows:

$\begin{array}{l}{\mathrm{\mathbb{Z}}}_{2^4}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2}\\ {\mathrm{\mathbb{Z}}}_{2^4}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\\ {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\end{array}$

Hence, the abelian groups of order 144 are:

$\left\{\begin{array}{l}{\mathrm{\mathbb{Z}}}_{2^4}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{3^2},\\ {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{3^2},{\mathrm{\mathbb{Z}}}_{2^4}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^3}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,\\ {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_2\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3,{\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_{2^2}\oplus {\mathrm{\mathbb{Z}}}_3\oplus {\mathrm{\mathbb{Z}}}_3\end{array}\right\}$