Question

$\text{Use the Fundamental Theorem of Finitely Generated Abelian Groups to answer the following:}$

$\text{a. } \text{Find all abelian groups, up to isomorphism, of order } $p^3$ \text{ where } $p$ \text{ is a prime.}$

$\text{b. } \text{Use part (a) with a suitable } $p$ \text{ to list all possible abelian groups that are isomorphic to } $(\mathbb{Z}_2 \times \mathbb{Z}_8)/\langle (1, 4) \rangle$.$

$\text{ From this list, identify the abelian group that is isomorphic to } $(\mathbb{Z}_2 \times \mathbb{Z}_8)/\langle (1,4)\rangle$.$

Answer 1

"a) By fundamental theorem group of order P^3 is isomorphic to z p^3 or z p^2 times z p or z p times z p Let a = z_2 times z_8/ < 1, 4 > note that 2 (1, 4) = (2, 8) = (0, 0) elementof z_2 times z_8 since