Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
134353
A group of order $108$ has a proper normal subgroup of order $\geq 6$.
44165
the image of normal subgroups
7575
Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
156319
Show that a group with $|G| = 33$ contains an element of order $3$
70373
Intersection of normal subgroups proof
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
80186
How to show that $K\leq S_4$ is a normal subgroup?
43651
Normal subgroups and factor groups
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n
12502
The center of a non-Abelian group of order 8
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
103423
Normal subgroup where G has order prime
141188
Groups of order $56$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
133298
Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.