How to show that $Z/12Z×Z/90Z×Z/25Z$ and $Z/100Z×Z/30Z×Z/9Z$ are isomorph?

Your proof using the primary decomposition is fine and systematic.

You can also use the Chinese remainder theorem to recombine the factors into the invariant factor decomposition:

$ G = \Bbb Z/12\Bbb Z \times \Bbb Z/90 \Bbb Z\times \Bbb Z/25 \Bbb Z \cong \Bbb Z/3\Bbb Z \times \Bbb Z/4\Bbb Z \times \Bbb Z/9 \Bbb Z \times \Bbb Z/10 \Bbb Z\times \Bbb Z/25 \Bbb Z \cong \Bbb Z/30\Bbb Z\times \Bbb Z/900 \Bbb Z $

$ H = \Bbb Z/100 \Bbb Z \times \Bbb Z/30\Bbb Z\times \Bbb Z/9\Bbb Z \cong \Bbb Z/30\Bbb Z\times \Bbb Z/100 \Bbb Z \times \Bbb Z/9\Bbb Z \cong \Bbb Z/30\Bbb Z\times \Bbb Z/900 \Bbb Z $

Tags:

Abstract Algebra

Group Theory

Group Isomorphism

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