---
tag: YB8V6TQDYE
kind: statement
role: theorem
name: Grushko theorem
msc: ["20E06", "20F05"]
uses: [TM7QS7P7X9, 0KTW26C80J, 082DWWXMRJ, 0JYTG5K9Y4, 01RVCMSN5N]
about: [TM7QS7P7X9, 082DWWXMRJ]
sources:
  - url: https://en.wikipedia.org/wiki/Grushko_theorem
same_as:
  wikidata: Q17019684
  wikipedia: Grushko theorem
---

## Statement

Let $A$ and $B$ be groups, and let $G = A * B$ be their [[free product|TM7QS7P7X9]].
For a [[finitely generated group|082DWWXMRJ]] $X$, let $\rank X$ be the least number of elements that generate $X$.

1. Let $F$ be a [[free group|0KTW26C80J]] of finite rank, and let $\varphi \colon F \to G$ be a [[surjective|01RVCMSN5N]] [[homomorphism|0JYTG5K9Y4]]. Then $F = F_1 * F_2$ for some subgroups $F_1$ and $F_2$ of $F$ with $\varphi(F_1) = A$ and $\varphi(F_2) = B$.
2. If $A$ and $B$ are finitely generated, then $\rank(A * B) = \rank A + \rank B$.

## Remarks

Part 2 follows from part 1.
If $G$ is generated by $n$ elements, take $F$ free of rank $n$ and $\varphi$ onto $G$.
Then $F_1$ and $F_2$ are free factors of $F$, so their ranks add up to $n$, and they map onto $A$ and $B$, so $\rank A + \rank B \le n$.
The other inequality is clear, since generators of $A$ and of $B$ together generate $A * B$.
Proofs of part 1 use Stallings foldings or [[Bass–Serre theory|6M731X66RD]].

Example: $\Z/2 * \Z/3$ has rank $2$, and $\Z^2 * \Z$ has rank $3$.

The free product cannot be replaced by the direct product.
The groups $\Z/2$ and $\Z/3$ have rank $1$ each, but $\Z/2 \times \Z/3 \cong \Z/6$ is cyclic, so its rank is $1$, not $2$.

## Explanations

### L1 human

The rank of a group is the smallest number of elements that generate it.
The theorem says that for a free product, the ranks just add.
No clever choice of generators can save anything when two groups are glued with no relations between them.

### L2 human

The first part says that every way of mapping a free group onto $A * B$ splits along a free product decomposition of the free group.
The rank formula is the usual form of the theorem.
It implies that every nontrivial finitely generated group is a free product of finitely many freely indecomposable groups.
Here a group is freely indecomposable if it is nontrivial and not a free product of two nontrivial groups.
The decomposition stops because each split lowers the ranks of the factors.
It fits with the [[Kurosh subgroup theorem|RY8K5RQJSR]] as one of the basic facts about free products.