# Free group universal property

The uni­ver­sal prop­erty of the free group ba­si­cally tells us that “the defi­ni­tion of the free group doesn’t de­pend (up to iso­mor­phism) on the ex­act de­tails of the set $$X$$ we picked; only on its car­di­nal­ity”, which is morally a very use­ful thing to know. You may skip down to the next sub­head­ing if you might be scared of cat­e­gory the­ory, but the prop­erty it­self doesn’t need cat­e­gory the­ory and is helpful.

The uni­ver­sal prop­erty is the tech­ni­cal cat­e­gory-the­o­retic fact that the free-group func­tor is left ad­joint to the for­get­ful func­tor, and it is not so im­me­di­ately use­ful as the other more con­crete prop­er­ties on this page, but it is ex­ceed­ingly im­por­tant in cat­e­gory the­ory as a very nat­u­ral ex­am­ple of a pair of ad­joint func­tors and as an ex­am­ple for the gen­eral ad­joint func­tor the­o­rem.

# State­ment and explanation

The uni­ver­sal prop­erty which char­ac­ter­ises the free group is:

The free group $$FX$$ on the set $$X$$ is the group, unique up to iso­mor­phism, such that for any group $$G$$ and any func­tion of sets $$f: X \to G$$ noteHere we’re slightly abus­ing no­ta­tion: we’ve writ­ten $$G$$ for the un­der­ly­ing set of the group $$G$$ here., there is a unique group ho­mo­mor­phism $$\overline{f}: FX \to G$$ such that $$\overline{f}(\rho_{a_1} \rho_{a_2} \dots \rho_{a_n}) = f(a_1) \cdot f(a_2) \cdot \dots \cdot f(a_n)$$.

This looks very opaque at first sight, but what it says is that $$FX$$ is the unique group such that:

Given any tar­get group $$G$$, we can ex­tend any map $$f: X \to G$$ to a unique ho­mo­mor­phism $$FX \to G$$, in the sense that when­ever we’re given the image of each gen­er­a­tor (that is, mem­ber of $$X$$) by $$f$$, the laws of a group ho­mo­mor­phism force ex­actly where ev­ery other el­e­ment of $$FX$$ must go. That is, we can spec­ify ho­mo­mor­phisms from $$FX$$ by spec­i­fy­ing where the gen­er­a­tors go, and more­over, ev­ery pos­si­ble such speci­fi­ca­tion does in­deed cor­re­spond to a ho­mo­mor­phism.

# Why is this a non-triv­ial prop­erty?

Con­sider the cyclic group $$C_3$$ with three el­e­ments; say $$\{ e, a, b\}$$ with $$e$$ the iden­tity and $$a + a = b$$, $$a+b = e = b+a$$, and $$b+b = a$$. Then this group is gen­er­ated by the el­e­ment $$a$$, be­cause $$a=a$$, $$a+a = b$$, and $$a+a+a = e$$. Let us pick $$G = (\mathbb{Z}, +)$$. We’ll try and define a map $$f: C_3 \to \mathbb{Z}$$ by $$a \mapsto 1$$.

If $$C_3$$ had the uni­ver­sal prop­erty of the free group on $$\{ e, a, b\}$$, then we would be able to find a ho­mo­mor­phism $$\overline{f}: C_3 \to \mathbb{Z}$$, such that $$\overline{f}(a) = 1$$ (that is, mimick­ing the ac­tion of the set-func­tion $$f$$). But in fact, no such ho­mo­mor­phism can ex­ist, be­cause if $$\overline{f}$$ were such a ho­mo­mor­phism, then $$\overline{f}(e) = \overline{f}(a+a+a) = 1+1+1 = 3$$ so $$\overline{f}(e) = 3$$, which con­tra­dicts that the image of the iden­tity un­der a group ho­mo­mor­phism is the iden­tity.

In essence, $$C_3$$ “has ex­tra re­la­tions” (namely that $$a+a+a = e$$) which the free group doesn’t have, and which can thwart the at­tempt to define $$\overline{f}$$; this is re­flected in the fact that $$C_3$$ fails to have the uni­ver­sal prop­erty.

A proof of the uni­ver­sal prop­erty may be found el­se­where.

Parents:

• Free group

The free group is “the purest way to make a group con­tain­ing a given set”.

• Group

The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.