Students will learn what abelian groups are and how to prove whether an operation over a set forms an abelian group. Students will also learn what the trivial group is, why it's a group, and why it's abelian. Here's a video covering these aspects of the trivial group.
Abelian groups
Lesson
Practice
Let \(G\) be a group and show that if \((ab)^2 = a^2b^2\) for all \(a, b \in G\), then \(G\) is abelian.
Let \(G\) be a group such that every element of \(G\) is equal to its own inverse. Show that \(G\) is abelian.
Prove that \(G\) is abelian if and only if \((ab)^{-1} = a^{-1}b^{-1}\) for all \(a, b \in G.\)