Let $a$ be a p-cycle in $S_p$, and let $b$ be a transposition in $S_p$. Show $S_p$ is generated by $a$ and $b$.
14139
$(12)$ and $(123\dots n)$ are generators of $S_n$
102234
Show $a^p \equiv a \mod 2p$
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
25713
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
4555
How to write permutations as product of disjoint cycles and transpositions
158355
If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$
13771
Why are two permutations conjugate iff they have the same cycle structure?
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
156247
Question about permutations and cycles (Groups and Symmetries)
155372
Find the number of permutations in $S_n$ containing fixed elements in one cycle
19026
What do all the $k$-cycles in $S_n$ generate?
108237
Show $a^p \equiv b^p \mod p^2$
133270
Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
82578
Show that if $G$ is $2-$connected, then for any two vertices $u$ and $v$ there exists a cycle $C$ such that $u, v \in V (C)$
143293
Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?
96680
Generators of symmetric group $S_n$