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Problem (Which Groups Of Order 60 Are Isomorphic)

Prove or disprove each of the following. Either build an isomorphism, or show that no such isomorphism exists.

  1. $\mathbb{Z}_4\oplus Z_{15}\approx \mathbb{Z}_{6}\oplus \mathbb{Z}_{10}$
  2. $\mathbb{Z}_4\oplus \mathbb{Z}_{15}\approx \mathbb{Z}_{20}\oplus \mathbb{Z}_{3}$
  3. $D_{20}\oplus \mathbb{Z}_{3}\approx D_{60}$ (Recall that $D_{2n}$ is the automorphisms of a regular $n$-gon.)
  4. $D_{20}\oplus \mathbb{Z}_{3}\approx \mathbb{Z}_{12}\oplus \mathbb{Z}_5$

Click to see a hint.

Consider orders of elements. What's the largest possible order in each group? How many elements of order 2 does each group have? If you find a mismatch between groups, they cannot be isomorphic.

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