If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
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 $G\cong M \times N$.
48585
Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
155219
Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
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 $G$
20386
Any subgroup of index $p$ in a $p$-group is normal.
97928
Normal subgroup of a normal subgroup
37971
Prove that the center of a group is a normal subgroup
14255
Commutator subgroup and cyclic normal subgroup
16804
Prove whether or not a subgroup is normal in the dihedral group $D_4$
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
109236
Subgroups containing kernel of group morphism to an abelian group are normal.
28108
Proving the kernel is a normal subgroup.