If every Sylow subgroup of a finite group $G$ is normal for every prime $p$, then $G$ is the direct product of its Sylow groups.
11957
G is the product of its Sylow subgroups
78553
A simple finite group $G$ with $n$ p-Sylows is isomorphic to a subgroup of $\mathbb A_n$
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
43651
Normal subgroups and factor groups
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
146278
Automorphism group of direct product of groups
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
55178
When there is a unique Sylow p-subgroup it is normal, does this mean that non-unique p-subgroups are not normal?
29736
Sylow Subgroups of a Dihedral Group
138406
If N and every subgroup of N is normal in G then G/N is abelian .
19352
Normal subgroup of index m contains every element of order n where m and n are relatively prime
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.
113357
Groups of order $pqr$ and their normal subgroups
65926
Group of order $1575$ having a normal sylow $3$ subgroup is abelian.
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$