Counterexample to $g \circ f$ one-to-one $\implies$ $g$, $f$ one-to-one
32127
Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective
151047
a non trivial example: $G/H$ is isomorphic to $G$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
104757
If $f \circ g$ is surjective, then f is surjective
113925
Normal Subgroup Counterexample
13150
If the derivative of $f$ is never zero, then $f$ is one-to-one
134703
Let $X$ be a connected space, $f:X\to X$ continuous involution, $g:X\to\mathbb{R}$ continuous. Prove that $\exists$ $x\in X$ such that $g(x)=g(f(x))$.
9204
Counterexamples to the avoidance lemma for arbitrary ideals
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
12455
$|G|>2$ implies $G$ has non trivial automorphism
70912
If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
153100
A counterexample to "$k\mid n$ if and only if $k\mid n^2$"
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
101773
Let $g : A \to B$ and $f : B \to C$ be functions. Let $f \circ g$ be onto. Are both $f$ and $g$ necessarily onto?
58642
If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?
82935
Prove that f∈Θ(g)if and only ifO(f) =O(g).
73869
$\mathcal{C}_1 \subseteq \mathcal{C}_2 \implies \sigma( \mathcal{C}_1) \subseteq \sigma( \mathcal{C}_2) $