If $M$ and $N/M$ are finitely generated, then so is $N$.
9806
$M$ finitely generated if submodule and quotient are finitely generated.
153100
A counterexample to "$k\mid n$ if and only if $k\mid n^2$"
134365
When is this true: If $m^2$ is a multiple of N then m is a multiple of N
76527
If $x_{m+n} \le x_n+x_m$, then $\lim x_n/n$ exists and is equal to $\inf x_n/n$
148413
Proving $S_n$ is generated by the set $\{(1\text{ }2),(2\text{ }3),\cdots,(n-1\text{ }n)\}$ of $n-1$ transpositions.
1712
$ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.
2287
If a set is linearly independent, a finite subset of that set is also linearly independent
128437
How to prove that a finite set is countable
67295
Please check my proof for $N$ is infinite
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
33810
Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
49709
Suppose $X$ is infinite and $A$ is a finite subset of $X$. Then $X$ and $X \setminus A$ are equinumerous
82778
If a measure is semifinite, then there are sets of arbitrarily large but finite measure
123847
Cardinality of $GL_n(K)$ when $K$ is finite
37012
The number of summands $\phi(n)$
59459
Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality