Determine the isomorphism class of the finite Abelian group

Determine the isomorphism class of the finite Abelian group U(56). Justify your answer.

Solution

|G| = 56 = 23 · 7.

We know that G = G(2) × G(7) where |G(2)| = 23 and |G(7)| = 7. Therefore the possible exponents for elementary divisors are given by the following lists: G(2) G(7) 3 1 2, 1 1, 1, 1 Therefore G must be isomorphic to one of the following groups: Z23 × Z71 = Z8 × Z7 Z22 × Z21 × Z71 = Z4 × Z2 × Z7 Z21 × Z21 × Z21 × Z71 = Z2 × Z2 × Z2 × Z7