Here, I found the following two exercises:
Problem IV. Let $G$ be an abstract group and let $a$ and $b$ be elements of $G$. Show that the map $L_a$ : $G \to G$ defined by $L_a(x) := a\star x$ is a bijection. You may find it useful to recall that, given $a$ and $b$ in $G$, the equation $a \star x = b$ has a unique solution in $G$.
Problem V. Let $G$ be an abstract group with the property that for every $g ∈ G$ we have $g \star g = e$. Show that $G$ is abelian, i.e., that for every $a$ and $b$ in $G$ we have $a \star b = b \star a$.
Now I wonder: What's an "abstract group"? I know what a group is. How is the term "abstract group" defined and why doesn't one just say "group"?