If $gHg^{-1} \subset H$, must we have $g^{-1}Hg \subset H$?
41085
Conjugate subgroup strictly contained in the initial subgroup?
93666
Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
108664
Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
81116
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
10024
How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
36631
Is $[0,1] \cap \Bbb Q$ a compact subset of $\Bbb Q$?
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
40383
Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$