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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 $\Bbb{Z}/2\Bbb{Z}$?

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Identical question math.stackexchange.com/questions/590651/… –  Tobias Kildetoft Dec 3 '13 at 9:17

1 Answer 1

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?

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