---
tag: 6M9W8D7Z7T
kind: proof
name: Proof that G is abelian if G modulo its centre is cyclic
msc: [20A05]
proves: 0AH9Q6YNMT
uses: [0S7Y9Q990Q, 0YYWCNTPWR, 0B85Q41RP6, 0YT586TKXF, 05FD317V2W, 00DRK90Z6K, 08FKQD7NJ3, 0HV9Y7WG5R, 071DDFVG4J]
sources:
  - url: https://proofwiki.org/wiki/Quotient_of_Group_by_Center_Cyclic_implies_Abelian
---

## Proof

We write each element of $G$ as a power of one element times a central element.

1. Let $G$ be a group with identity $e$ and [[centre|0S7Y9Q990Q]] $Z = Z(G)$. Assume that the [[quotient group|0YYWCNTPWR]] $G/Z$ is [[cyclic|0B85Q41RP6]], with generator $tZ$ for some $t \in G$.
2. The projection $\pi \colon G \to G/Z$, $\pi(x) = xZ$, satisfies $\pi(xy) = (xy)Z = (xZ)(yZ) = \pi(x) \pi(y)$. So $\pi$ is a [[group homomorphism|0HV9Y7WG5R]].
3. For $k \ge 0$, $\pi(t^k) = \pi(t)^k$ by induction on $k$. The case $k = 0$ holds because [[homomorphisms preserve the identity|08FKQD7NJ3]], and $\pi(t^{k+1}) = \pi(t^k) \pi(t)$.
4. For $k > 0$, $\pi(t^{-k}) = \pi((t^{-1})^k) = \pi(t^{-1})^k = (\pi(t)^{-1})^k = \pi(t)^{-k}$, since homomorphisms preserve inverses.
5. By steps 3 and 4, $(tZ)^k = t^k Z$ for each integer $k$.
6. Let $x \in G$. By the [[definition of $\langle tZ \rangle$|0YT586TKXF]] and step 5, $xZ = t^m Z$ for some integer $m$.
7. Since $e \in Z$, $x = xe$ lies in the [[left coset|05FD317V2W]] $xZ = t^m Z$. So $x = t^m z_1$ for some $z_1 \in Z$.
8. Let $x, y \in G$. By step 7, $x = t^m z_1$ and $y = t^n z_2$ with integers $m, n$ and $z_1, z_2 \in Z$.
9. Since $z_1$ commutes with every element, $xy = t^m z_1 t^n z_2 = t^m t^n z_1 z_2$. By the [[laws of exponents|00DRK90Z6K]], $xy = t^{m+n} z_1 z_2$.
10. In the same way, $yx = t^n z_2 t^m z_1 = t^n t^m z_2 z_1 = t^{m+n} z_2 z_1$.
11. Since $z_1 \in Z$, $z_1 z_2 = z_2 z_1$. By steps 9 and 10, $xy = yx$.
12. Step 11 holds for all $x, y \in G$. So $G$ is [[abelian|071DDFVG4J]].

## Idea

Modulo the centre, every element is a power of $t$, so each element is $t^m$ times a central element. Powers of the same $t$ commute with each other, and central elements commute with everything. So any two elements commute.