Abstract Algebra: Theory and Applications

Determine the order of $(1,2)$ in ${\mathbb{Z}}_4\times {\mathbb{Z}}_8\text{.}$

List three properties of a group that are preserved by an isomorphism.

Find a group isomorphic to ${\mathbb{Z}}_{15}$ that is an external direct product of two non-trivial groups.

Explain why we can now say “the infinite cyclic group”?

Compare and contrast external direct products and internal direct products.