If cancellation laws hold, then a finite semi-group is a group
6755
A finite, cancellative semigroup is a group
14097
A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
96614
Finite number of subgroups $\Rightarrow$  finite group
88344
Order of a group and subsequent subgroups
142421
Conjugacy Classes in Finite Groups
109034
Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite?
35124
Can every non-simple group $G$ be written as a semidirect product?
43750
Subgroups of finite solvable groups. Solvable?
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$
85379
Semigroup analogue to the classification of finite simple groups?
152904
Groups with Same Number Of Elements and Subgroups
116707
Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.
5089
An element of a group has the same order as its inverse
139611
Can a normal subgroup of a finite nonabelian group be nonabelian?
151940
Cancellation law for direct sums
66510
A group is abelian if and only if the center of the group is all the group