Question

Show that subgroup and factor group of nilpotent group is nilpotent?

Answer 1

Answer on Question #52631 – Math – Abstract Algebra

Show that subgroup and factor group of nilpotent group is nilpotent?

Proof

We use the idea of a central series for a group. The following are equivalent formulations:

- A nilpotent group is one that has a central series of finite length.

- A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps.

- A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps.

Suppose $Z_{n}(G) = G$ and let $H \leq G$. For each $r$, $Z_{r}(H) \geq Z_{r}(G) \cap H$ and so $Z_{n}(H) = H$.

Suppose now that $H$ is normal in $G$. For each $r$, $Z_{r}(G)H / H \geq Z_{r}(G / H)$ and so

$Z_{n}(G / H)$ is trivial for any $n$ by the method of mathematical induction.

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