A subgroup with an index of 2 is normal to the group.
75568
Subgroup of index $2$ is Normal
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
111375
Normality of subgroups and Sylow groups under homomorphisms...
116786
normal subgroups of Alternating group
20386
Any subgroup of index $p$ in a $p$-group is normal.
113925
Normal Subgroup Counterexample
38095
The Commutator Subgroup is Normal.
30793
Intuition behind normal subgroups
57227
Maximal normal $\pi$-subgroups and torsion subgroups
22444
Normal subgroups of dihedral groups
13523
Are normal subgroups transitive?
61373
A normal subgroup that is not a characteristic
63780
to show characteristic subgroup is normal in G
109060
Group containing no subgroup of index 2
49122
Why commutator subgroup is normal to G?
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
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 $G=HK$?