Priority of the 3 axioms of groups 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.
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.