There are several claims in the literature that there are no injective groups (with more than one element), but I have not found a proof. For example, Mac Lane claims in his Duality from groups paper in the 1950 Bull. A.M.S. that Baer showed him an elegant proof, but gives no hint of what it might be. Does anyone know of an actual published proof?

8$\begingroup$ Google gives zimmer.csufresno.edu/~mnogin/talks/regAMSapril2004.pdf which may not be formally published, but is definitely short. $\endgroup$– user2035Jun 21, 2012 at 13:52

1$\begingroup$ And sweet, definitely short and sweet. $\endgroup$– Lee MosherJun 21, 2012 at 14:06

2$\begingroup$ That's a nice proof! Since it is short, it could be copied as an answer below (with proper attribution) so it is easier to find and cite... $\endgroup$– François G. DoraisJun 21, 2012 at 14:16

$\begingroup$ About Maria Voloshina proof, that injectivity of $f: F[a, b]\to F[c, d]:a \mapsto c,\ b\mapsto d\cdot c\cdot d^{1} $ follow from $\forall n:P(n)$, where $P(n)$:: the (reducted) word f(w) end with $c$ or by $d^{−1}$ for each (reducted) word $w\in F[a,b]$ of lenght $n$ ($n>0$). And $\forall n>0:P(n)$ follow easly by induction. $\endgroup$– Buschi SergioJun 21, 2012 at 15:48

6$\begingroup$ This question has appeared in MSE a few months ago: math.stackexchange.com/questions/102820/… $\endgroup$– Bruno StonekJun 21, 2012 at 16:03
3 Answers
The argument in the slides by Maria Nogin (née Voloshina) linked to by afortiori in a comment was published as:
Maria Nogin, A short proof of Eilenberg and Moore’s theorem, Central European Journal Of Mathematics Volume 5, Number 1 (2007), 201–204. Also available on her homepage.
Added: The above paper was also mentioned in Jonas Meyer's answer to the same question on math.SE. As Steve D. points out in a comment there, the result appears as Exercise 7 on page 9 of D.L. Johnson's Topics in the Theory of Group Presentations, Cambridge University Press 1980:
The argument by Eilenberg and Moore appears on pages 21/22 of Foundations of Relative Homological Algebra, Memoirs of the AMS, Volume 55 (1965). Here's a scan of the relevant passage for the convenience of the readers:

$\begingroup$ I forgot: EilenbergMoore refer to the paper by R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich, Studia Math. 5 (1934), 1517 which is available on the home page of the Polish virtual library of science: pldml.icm.edu.pl. Direct link to the paper: matwbn.icm.edu.pl/ksiazki/sm/sm5/sm512.pdf $\endgroup$ Jun 21, 2012 at 19:22

$\begingroup$ I just wanted to say that the proof attributed to Maria Nogin uses more about homotopy than I wanted my readers to have to know. And there is something wrong with the inductive argument that that map is 11 since clearly the image of the length 1 word $a^{1}$ does not end in either $c$ or $d^{1}$. I guess the argument of EilenbergMoore may be the easiest. Now I have to track down the reference to Baer. Thanks to all who answered. $\endgroup$ Jun 21, 2012 at 20:41

$\begingroup$ @Michael: I'm not sure what inductive argument you are referring to here. Nogin's proof uses only that the inclusion of a subgraph into the full graph is $\pi_1$injective, and that covering maps are $\pi_1$injective. The subgraph and covering map to which these facts are applied are very concrete in her paper. $\endgroup$ Jun 21, 2012 at 22:17

$\begingroup$ See the comment added by Sergio Buschi added to #2. I don't know enough homotopy theory to see why that map is injective on $\pi_1$. Actually, I should have recalled that EilenbergMoore had it. I was very familiar with it back when it first came out. But it was kind of overtaken by Beck's thesis and other things and disappeared from my memory. $\endgroup$ Jun 22, 2012 at 18:23
Let $G$ be a nontrivial injective group and $g \in G$ nontrivial. From category theory, $G \times G$ is injective as well. But, now $G \times G$ embeds into a group $H$, where $(g,e)$ and $(e,g)$ are conjugate using an HNNextension, or just embedding into the group $H:=(G \times G) \rtimes \mathbb Z/2\mathbb Z$. Since $(g,e)$ and $(e,g)$ are not conjugate in $G \times G$, $H$ cannot split back to $G \times G$. This is a contradiction.
Does this work? I want to use the following statement, which I believe is true.
Let $S$ be an infinite set. Let $\Pi(S)$ be the group of bijections $S\to S$. Let $N$ be the normal subgroup consisting of those bijections whose support is smaller (in cardinality) than $S$. Then every proper normal subgroup of $\Pi(S)$ is contained in $N$.
Now suppose that $G$ is a nontrivial injective group. Certainly it is infinite. Let $S$ be the set of elements of $G$. $S$ is isomorphic to a subgroup of $\Pi(S)$. Being injective, it is therefore isomorphic to a quotient of $\Pi(S)$, and is therefore at least as big as $\Pi(S)/N$, which is as big as $2^S$, contradiction.

$\begingroup$ (crossposted with the simpler and more sensible answer given in the comments) $\endgroup$ Jun 21, 2012 at 15:14