If $H$ is a subgroup of $G$ of finite index $n$, then under what condition $g^n\in H$ for all $g\in G$
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
64408
Subgroup of $\mathbb{Q}$ with finite index
142228
Proof that a subset closed under group operation of a finite group is a subgroup
75568
Subgroup of index $2$ is Normal
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
39165
Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
46121
Does the intersection of two finite index subgroups have finite index?
58863
Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
19352
Normal subgroup of index m contains every element of order n where m and n are relatively prime
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
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 $G$
155502
Index of a maximal subgroup in a finite group
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.