Finite group with cyclic Sylow-2 group has a normal subgroup of index 2.
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
97928
Normal subgroup of a normal subgroup
7108
Normal subgroup of prime index
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
152954
Extensions of finite cyclic groups
99114
Subgroups of index 3 are normal
15341
Non-abelian finite group in which more than half of the elements have order $2$
142938
Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$
155502
Index of a maximal subgroup in a finite group
2731
Groups of even order have an element whose order is 2.
28210
A finite group with exactly $2$ conjugacy classes isomorphic to $\mathbb{Z}_2$
8623
Finite subgroups of the multiplicative group of a field are cyclic
94999
A subgroup with an index of 2 is normal to the group.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
2254
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
58914
Any group of index 2 is normal