Use Hall's Theorem to show that if $|N(S)| \ge |S| + 1$, $G$ has a perfect matching which contains an edge $e$.
140114
Using Halls Theorem to show Graph contains a perfect matching containing any edge.
96329
Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$
152987
Dilworth's Theorem Implies Konig's Theorem
6429
prove that a connected graph with $n$ vertices has at least $n-1$ edges
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
73253
Question About equal signs in Theorems
41641
$(n-7)(n+7)=$ some perfect square
114777
Using Rouché's theorem to bound zeroes of partial sums of $e^z$
101832
Proof that sum of first $n$ cubes is always a perfect square
79730
$G$ is a planar, connected graph. Let $n$ be number of vertices, $m$ number of edges and $f$ number of faces. Prove that dual graph $G*$ has $f(G*)=n$
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
101179
Partition of a set and Hall's theorem
33316
Show that if $G$ is simple a graph with $n$ vertices and 􏰈the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
130081
Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
71154
Why a complete graph has $\frac{n(n-1)}{2}$ edges?
156319
Show that a group with $|G| = 33$ contains an element of order $3$