Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$.
5226
Prove that a finite semigroup G with identity is a group if and only if G contains only one idempotent
132127
Proving a direct product of groups is a group
17997
Right identity and Right inverse implies a group
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
6153
Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
101345
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
26398
Group of even order contains an element of order 2
13631
Showing group with $p^2$ elements is Abelian
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
29538
Does a non-abelian semigroup without identity exist?
121765
Semigroups and solutions of equation
13355
Product of elements of a finite abelian group
121077
Product of all elements in finite group
19683
Is a semigroup $G$ with left identity and right inverses a group?
156549
Prove that there is no group $G$ s.t. $\operatorname{Aut}(G)=\mathbb{Q}$
102859
Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.