If $H$ is a subgroup of $S_n$ and if H is not contained in $A_n$ prove that precisely one half of the elements of H are even permutations.
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
3439
Other proofs that subgroups of $A_5$ have order at most 12
44480
Let $\sigma :\{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ be permutations such that $ \sigma^{-1}(j) \leq \sigma(j)~\forall j, 1 \leq j \leq 5$.
7575
Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
142228
Proof that a subset closed under group operation of a finite group is a subgroup
97833
Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$
104549
When $d \mid n$, the number of elements of order $d$ in a cyclic subgroup of order $n$ is $\phi (d)$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
151130
When is the product of $n$ subgroups a subgroup?
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
70373
Intersection of normal subgroups proof
41231
Subgroups of (Z,+) of order n
43058
If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
96329
Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$