Prove that $[G:K]=[G:H][H:K] $where $K\le H \le G$
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
63447
Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13)
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
28722
Prove that $H$ is a abelian subgroup of odd order
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
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 $G$
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
147398
If every element of $H$ and $G/H$ is a square, then prove that so is every element of $G$.
39708
Proving $(A\le B)\vee (B\le A)$ for sets $A$ and $B$
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
79367
Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.
151047
a non trivial example: $G/H$ is isomorphic to $G$
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$