How to show that if $H \leq G$ and $H$ is the only subgroup of $G$ of order n than $ H\trianglelefteq G$
71819
Prove that $H$ is normal.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
117899
At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
98474
Show the existence of a certain subgroup of F2
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
88344
Order of a group and subsequent subgroups
89508
Existence of normal subgroups for a group of order $36$
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
41231
Subgroups of (Z,+) of order n
109053
Show that a finite cyclic group of order $n$ has exactly one subgroup of order $d$ dividing $n$ and these are all the subgroups it has
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
76943
Existence of subgroup of order six in $A_4$