Suppose $gH = Hg, \forall g \in G$. does this imply that $g^{-1}Hg \subseteq H, \forall g \in G$?
125244
Equivalent definitions of normal subgroups
110672
Is it true that $aH = bH$ iff $ab^{-1} \in H$
35353
Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
93157
If $A \subseteq B$, does $\mbox{dist}(x,\partial A) \le \mbox{dist}(x, \partial B)$ hold for all $x \in A$
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
21217
$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
79826
$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$
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
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
103216
The set $\{g^2 | g \in G\}$ in a group $G$
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?