If $K\leq H\leq G$ (not necessarily finite groups). Then prove that $[G:K]=[G:H]\cdot [H:K]$
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
129452
Groups $G,H$ with surjective homomorphisms that are quotient of each other
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
98542
Prove that $H$ is a subgroup of an abelian group $G$
80186
How to show that $K\leq S_4$ is a normal subgroup?
146760
Prove that H is a subgroup of G
58033
$G\times H\cong G$ with $H$ non-trivial
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$