If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
15276
$G \times H \cong G \times K$ , then $ K \cong H$
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
65744
How can I Prove that H=G or K=G
142228
Proof that a subset closed under group operation of a finite group is a subgroup
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
70373
Intersection of normal subgroups proof
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$