Group theory and generalizations Theorem

Universal property of the free group

Statement

Let $S$ be a set, let $F_S$ be the free group on $S$ , and let $G$ be a group. For each function $f \colon S \to G$ , there is exactly one group homomorphism $\varphi \colon F_S \to G$ with $\varphi(s) = f(s)$ for each $s \in S$ .

Remarks

Here $S$ is identified with the set of words of length $1$ in $F_S$ . The homomorphism is $\varphi(t_1 \cdots t_n) = f(t_1)^{\pm 1} \cdots f(t_n)^{\pm 1}$ , with the sign of each letter. This property characterises $F_S$ up to isomorphism, so some texts use it as the definition of a free group. If the image of $f$ generates $G$ , then $\varphi$ is surjective, so $G$ is a quotient of $F_S$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Group concept
Reduced word definition
Group homomorphism definition
Group isomorphism definition
Free group definition
Universal property of the free group statement

Where this leads

1 page builds on this one.

Von Dyck's theorem

Tag	`0CDWKDYWN6`
URI	`m2://0CDWKDYWN6`
Kind	statement, theorem
Writing	W0 stub
MSC	20E05
Status	This copy of the site has no build data. Run `m2 status 0CDWKDYWN6`.
Uses	Free group, Group homomorphism

Universal property of the free group

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by