If the cardinality of a subgroup is half the cardinality of the original group, is the subgroup normal?
75568
Subgroup of index $2$ is Normal
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
35602
If a subgroup has smallest prime index, then it is normal
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
104505
Is every normal subgroup the kernel of some endomorphism?
102912
Subgroups of finitely generated groups are not necessarily finitely generated
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
103423
Normal subgroup where G has order prime
14666
What are the non-trivial normal subgroups of $O(3)$?
75590
How many normal subgroups does a non-abelian group $G$ of order $ 21$
125244
Equivalent definitions of normal subgroups
111375
Normality of subgroups and Sylow groups under homomorphisms...
49666
Generalization of index 2 subgroups are normal
7108
Normal subgroup of prime index
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
91874
Existence of finite indexed normal subgroup for a given finite indexed subgroup.