Showing H is a normal subgroup of G
94999
A subgroup with an index of 2 is normal to the group.
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
71819
Prove that $H$ is normal.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
115204
Show that SO(n) is a normal subgroup of O(n)
11957
G is the product of its Sylow subgroups
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
113357
Groups of order $pqr$ and their normal subgroups
65744
How can I Prove that H=G or K=G
80186
How to show that $K\leq S_4$ is a normal subgroup?
70373
Intersection of normal subgroups proof
16804
Prove whether or not a subgroup is normal in the dihedral group $D_4$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.