Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$.
5226
Prove that a finite semigroup G with identity is a group if and only if G contains only one idempotent
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
145406
Prove every element of $G$ has finite order.
13631
Showing group with $p^2$ elements is Abelian
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
11003
Proving that $G/N$ is an abelian group
7536
Is there an idempotent element in a finite semigroup?
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
152191
Why is a monoid with right identity and left inverse not necessarily a group?
121077
Product of all elements in finite group
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
145254
$G$ be a non-nilpotent and supersoluble finite group.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.