Prove whether the subgroup of group is always normal
57883
Example of a Subgroup That Is Not Normal
139096
$|AxA|$ is a constant implies normal subgroup
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
20386
Any subgroup of index $p$ in a $p$-group is normal.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
104505
Is every normal subgroup the kernel of some endomorphism?
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
132704
If G is not commutative, then is there always a subgroup that is not a normal subgroup?
51747
Showing that the intersection of all subgroups of order $n$ is normal
116786
normal subgroups of Alternating group
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
142228
Proof that a subset closed under group operation of a finite group is a subgroup
133298
Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.