$H, K$ normal subgroups of a finite group $G$ with $G=HK$, $P$ is a Sylow $p$-subgroup of $G$, then $P = (P\cap H)(P\cap K)$
61235
Sylow $p$-subgroup of a direct product is product of Sylow $p$-subgroups of factors
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
118337
Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
154398
Show that H is a subgroup of G
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
11731
Normal subgroups of $S_4$
58251
To show every sylow p-subgroup is normal in G
42096
Direct product of Sylow subgroups
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
29736
Sylow Subgroups of a Dihedral Group
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
45580
Intersection of conjugate subgroups is normal
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
70373
Intersection of normal subgroups proof