Abstract Algebra: Cyclic and Abelian Groups

Show that a group G is abelian if for all elements a and b in G, a * b = b * a.

Demonstrate that the group of symmetries of a square in 3D is not abelian by finding elements a and b such that a * b ≠ b * a.

In an abelian group, demonstrate that left and right cosets must be the same by using the group Z8 and the subgroup {0, 4}. Compute left cosets and right cosets for a specific element and confirm their equality.

Are all cyclic groups Abelian? And are all Abelian groups cyclic?

Find a counterexample of a cyclic group that is not Abelian or an Abelian group that is not cyclic.

Every cyclic group is Abelian. Prove it.

Prove that every cyclic group is abelian by taking two arbitrary elements from the group and showing that they commute.

Determine the order of the direct sum $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and determine if this group is cyclic.

Show that the group $\mathbb{Z}_7 \times$ is a cyclic group. Find all its generators, proper subgroups, and the order of every element.