Question

find all finite abelian group of order less than or equal to 40 up to isomorphism.

Answer 1

Trivial group $G_1 = \{e\}$ is abelian group of order 1.

Groups of prime order less than 40 are cyclic and hence abelian, there is one and only one group upto isomorphism per prime $p, Z_p$ where $p$ equal to $2,3,5,7,11,13,17,19,23,29,31,37$.

Divide the composite number into its prime factor, such as factors gcd not equal to one. For example divide $Z_{20}$ as $Z_2\times Z_{10}$ both are abelian group of same order.

However if gcd is 1, then they are isomorphic. For example: $Z_4\times Z_5 \cong Z_{20}$

List of all abelian group of order less than or equal to 40 utpo isomorphism are:

$Z_1 \\ Z_2 \\ Z_3 \\ Z_4 \\ Z_2 \times Z_2 \\ Z_5 \\ Z_6 \\ Z_7 \\ Z_8 \\ Z_4 \times Z_2 \\ Z_2 \times Z_2 \times Z_2 \\ Z_9 \\ Z_3 \times Z_3 \\ Z_{10} \\ Z_{11} \\ Z_{12} \\ Z_6 \times Z_2 \\ Z_{13} \\ Z_{14} \\ Z_{15} \\ Z_{16} \\ Z_4 \times Z_4 \\ Z_8 \times Z_2 \\ Z_4 \times Z_2 \times Z_2 \\ Z_2 \times Z_2 \times Z_2 \times Z_2 \\ Z_{17} \\ Z_{18} \\ Z_6 \times Z_3 \\ Z_{19} \\ Z_{20} \\ Z_{10} \times Z_2 \\ Z_{21} \\ Z_{22} \\ Z_{23} \\ Z_{24} \\ Z_{12} \times Z_2 \\ Z_6 \times Z_2 \times Z_2 \\ Z_{25} \\ Z_5 \times Z_5 \\ Z_{26} \\ Z_{27} \\ Z_9 \times Z_3 \\ Z_3 \times Z_3 \times Z_3 \\ Z_{28} \\ Z_{14} \times Z_2 \\ Z_{29} \\ Z_{30} \\ Z_{31} \\ Z_{32} \\ Z_8 \times Z_4 \\ Z_{16} \times Z_2 \\ Z_4 \times Z_4 \times Z_2 \\ Z_8 \times Z_2 \times Z_2 \\ Z_4 \times Z_2 \times Z_2 \times Z_2 \\ Z_2 \times Z_2 \times Z_2 \times Z_2 \times Z_2 \\ Z_{33} \\ Z_{34} \\ Z_{35} \\ Z_{36} \\ Z_{18} \times Z_2 \\ Z_{12} \times Z_3 \\ Z_6 \times Z_6 \\ Z_{37} \\ Z_{38} \\ Z_{39}\\ Z_{40} \\ Z_{20} \times Z_2 \\ Z_{10} \times Z_2 \times Z_2.$