Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Find all groups $G$ with a normal subgroup $H\simeq\Bbb{Z}$ and such that $[G:H]=2$.

How do I classify such groups? One obvious example is $\Bbb{Z}/2\Bbb{Z}$?

share|cite|improve this question
Identical question… – Tobias Kildetoft Dec 3 '13 at 9:17

Here is the start to one approach to this problem.

Let $h$ be a generator $\mathbb Z\cong H \subset G$, and let $g\in G$ with $g\not\in H$. Then $g$ and $h$ generate $G$. Moreover, $g^2=h^a$ for some $a\in \mathbb N$ because $G/H \cong \mathbb Z /2 \mathbb Z$, and $ghg^{-1}=h^b$ for some $b\in \mathbb N$ because $H$ is normal. Therefore, the groups you are looking for are classified in some sense by the ordered pairs $(a,b)$. However, this is incomplete because

  1. Not every ordered pair needs to correspond to an actual group, and
  2. Different ordered pairs could correspond to the same group.

Can you fix these two problems?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.