Proof that f : G → H is a monomorphism if and only if ker(f)={e}.
54360
Injective implies trivial kernel.
101064
Showing the triviality of homomorphisms
103491
If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
85612
Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
106550
The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$
137233
$f$ is irreducible $\iff$ $G$ act transitively on the roots
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
100439
Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
109034
Does a finite $[E:K]_s$ imply that $[E_s:K]$ is finite?
11202
If $g \circ f$ is injective, so is $g$
104757
If $f \circ g$ is surjective, then f is surjective
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
6825
If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
72251
$g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective