Let $N$ normal in $G$ such that every subgroup of $N$ is normal in $G$ and $C_G(N) ⊆ N$. Prove that $G/N$ is abelian.
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
135039
${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
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 $[G:N]=p$
91772
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
71420
Prove that a group is abelian.
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
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 $G$
70373
Intersection of normal subgroups proof
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
101345
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
48393
Abelian groups of order n.
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.
58205
$ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(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 $G$.