For $n\geq2$, prove that $A_n$ is the only subgroup of $S_n$ with index two.
58863
Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
88718
A group is generated by two elements of order $2$ is infinite and non-abelian
49666
Generalization of index 2 subgroups are normal
33387
Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$
76943
Existence of subgroup of order six in $A_4$
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
91772
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
151130
When is the product of $n$ subgroups a subgroup?
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
137691
What are the self-normalizing subgroups of $S_n$?
42292
Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n
143314
Sylow subgroup of $S_{p^2}$