In my book about Abstract Algebra, it is stated that
A group $\langle G,*\rangle$ is a set $G$, closed under a binary operation $*$, such that these 3 axioms are satisfied:
- $g_1$: For all $a,b,c\in G$, $$(a*b)*c=a*(b*c) $$
- $g_2$: There is an element $e$ in $G$ such that for all $x\in G$, $$e*x=x*e=e$$
- $g_3$: Corresponding to $a\in G$, there is an element $a'$ such that $$a*a'=a'*a=e$$
The book stated that among the 6 orders of $g_1,g_2,g_3$, only $3$ are correct orders of defining a group. But I have no idea about it. Can someone give me some hint?
The author of your book is unnecessarly delving into what order of stating axioms are useful. A dictionary should simply list what are the words of that language.
Listing a cacaphony of letters that are not words of that language, even if logically correct, does not serve any useful purpose.