Suppose a finite set $G$ is closed under associative product and that both cancellation laws hold in $G$. Prove $G$ must be a group.
6755
A finite, cancellative semigroup is a group
142228
Proof that a subset closed under group operation of a finite group is a subgroup
150007
Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group.
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
128437
How to prove that a finite set is countable
35124
Can every non-simple group $G$ be written as a semidirect product?
104855
Prove that $N \subset Z(G)$
114431
Monoids and Subgroups
143457
Is product of two closed sets closed?
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
126927
$\mathbb Q$ is not the direct product of two non-trivial groups.
14097
A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$
140103
If product of two sets $A\times B$ is closed, are $A$ and $B$ closed?
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
61394
The product of finitely many cyclic groups is cyclic
50026
Product of compact and closed in topological group is closed
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.