$G$ be a finite group, if $H\lneq G$ and $|G| \nmid [G:H]!$. Prove $G$ is not simple
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
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 $ab\in H$.
118022
Show that a group of order $180$ is not simple.
12455
$|G|>2$ implies $G$ has non trivial automorphism
112974
The $h-$th power of every element in a finite group of order $h$ is the identity element of the group
98542
Prove that $H$ is a subgroup of an abelian group $G$
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
24181
Showing that $G$ is a group under an alternative operation.
27436
Does $G\cong G/H$ imply that $H$ is trivial?
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
87534
Finitely generated group which is not finitely presented
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?