Group axioms

The group axioms require closure and associativity of the binary operation, an identity element, and an inverse for every element.

An associative binary operation $\ast$ on a nonempty set $X$ makes $X$ a group if there is $e\in X$ with $x\ast e=x$ for every $x$, and every $x$ has some $z$ with $x\ast z=e$. No left identity element or left inverse element assumption is needed. First, an idempotent element of a semigroup $y$ satisfies $y=e$: choose $z$ with $y\ast z=e$ and compute $y=y\ast e=y\ast (y\ast z)=(y\ast y)\ast z=y\ast z=e$. If $x\ast z=e$, then $(z\ast x)\ast (z\ast x)=(z\ast (x\ast z))\ast x=(z\ast e)\ast x=z\ast x$, so $z\ast x=e$ by the previous observation. Finally, $e\ast x=(x\ast z)\ast x=x\ast (z\ast x)=x\ast e=x$. Thus all group axioms follow.

Suppose a nonempty semigroup has, for each $a$, elements $l_a,r_a$ satisfying $b{l}_{a}a=b$ and $a{r}_{a}b=b$ for every $b$. Then it is a group. For any fixed $a$, the element $e_R=l_{a}a$ is a right identity element, and $e_L=a{r}_a$ is a left identity element. Since $e_L{e}_R=e_L=e_R$, they coincide in a two-sided identity $e$. For each $a$, putting $b=e$ in the two assumed identities gives $l_{a}a=e$ and $a{r}_a=e$. Associativity then yields $l_a=l_{a}e=l_a(a{r}_a)=(l_{a}a)r_a=e{r}_a=r_a$, a two-sided inverse element. Thus the usual group axioms follow without an assumed identity or cancellation law.