If $H_{3}$ is subgroup of $G$ that satisfies $H_3 \subseteq H_1 \cup H_2$ and $H_3 \not\subset H_2$, then $H_3 \subseteq H_1$.
13059
If a group is the union of two subgroups, is one subgroup the group itself?
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
85677
Is there a free subgroup of rank 3 in $SO_3$?
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
151130
When is the product of $n$ subgroups a subgroup?
93666
Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
141265
Finding a normal and not normal subgroup of $S_3$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ 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 $G$.
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
154398
Show that H is a subgroup of G
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.