In this section you will get comfortable with the definition of a group by proving some basic properties. You will also witness some of the consequences and effects of a group being abelian.
Basic group theory proofs
Lesson
The identity element is unique.
The inverse of an element is unique.
Shoes and socks
\((ab)^{-1} = b^{-1}a^{-1}\)
Left cancellation law
\(ab = ac \longrightarrow b = c\)
Right cancellation law
\(ba = ca \longrightarrow b = c\)
Practice
Let \(G\) be a finite group of even order. Show that \(G\) has an element \(a \ne e\) such that \(a^2 = e.\)
Let \(G\) be a group and fix \(g_0 \in G\). Prove that \(f : G \longrightarrow G\) given by \(f(x) = xg_0\) is a bijection.