0

If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G.

Attepmt

Let G be a group. Let H be a Normal subgroup of G. Let $K=\langle a \rangle$ be a cyclic subgroup of G generated by $a\in G$. We shall have to show that H is normal in G. That is to prove that for $h\in H$, $ghg^{-1}\in G$ for all $g\in G$.

If the steps I have written, please help me to solve the problem.

Stefan4024
  • 36,357
user1942348
  • 4,256
  • In Clayton solution, http://math.stackexchange.com/questions/276957/if-every-cyclic-subgroup-of-g-is-normal-so-is-every-subgroup,

    why $g\langle k\rangle g^{-1}=\langle k\rangle$ ?

     – user1942348 Oct 07 '16 at 09:30
  • Because the cyclic group $\langle k\rangle $is normal in $G$. This is the definition of being normal. – Dietrich Burde Oct 07 '16 at 09:35

1 Answers1

1

Let $H$ be a subgroup of $G$, $h \in H$ and $g \in G$. You have to show that $g^ {-1}hg \in H$. Since $H$ is a subgroup, the cyclic group $\langle h \rangle$ generated by $h$, is a subgroup of $H$ and hence of $G$. Your premise tells us that $\langle h \rangle$ is normal, so $g^ {-1}hg \in \langle h \rangle$, since of course $h \in \langle h \rangle$. Hence $g^ {-1}hg=h^i$ for some $i$, and $h^i \in H$.

Nicky Hekster
  • 52,147