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In Wiki, under the item "category of groups", it states that the snake lemma fails in category of groups, however the nine lemma is valid. However, in the preface of the book " Mal'cev, protomodular, homological and semi-abelian categories ", it says "And category theory could not grasp either the conceptual foundations of the homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid and strongly meaningful in the category Gp of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way."

Does the snake lemma fail or not in the category of groups? Any counter example or reference concerning that?

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2 Answers 2

up vote 9 down vote accepted

The snake lemma doesn't make sense in the category of groups in general, because maps don't always have cokernels. If you assume that the three vertical maps in your snake lemma diagram do have cokernels, that is, that their images are normal subgroups of their targets, then the snake lemma is true with the same proof as in the abelian category case (replacing the $0$'s by $1$'s and the minus signs by inversions, as J. Polak pointed out).

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That confirms what I suspected when reading the question. The question is what one considers a cokernel and thus what formulation of the snake lemma one wants. If it is the categorical factorization property then it's the quotient modulo the normal subgroup generated by the image and I'm pretty sure that that's what Borceux-Bourn have in mind (I can't check this because Google won't let me read the relevant pages). –  Theo Buehler Jan 24 '11 at 22:51
    
Some years ago I tried to prove the Snake Lemma for groups, but it didn't work. The usual diagram chase actually used commutativity. But I don't remember ... –  Martin Brandenburg Jan 25 '11 at 7:53
    
I did the diagram chase myself. I didn't need to use commutativity, but I did need several times the fact the vertical maps (and the horizontal maps on the first line) have normal image. This nicely replaces commutativity in some arguments, when you use the fact that a normal subgroup has the same left classes and right classes. –  Alex Jan 25 '11 at 16:50
    
@Theo Buehler : It seems you are mentioning the useful references about semi-abelian categories Bourn, Dominique 3×3 lemma and protomodularity. (English summary) J. Algebra 236 (2001), no. 2, 778–795. and Borceux, Francis(B-UCL) A survey of semi-abelian categories. (English summary) Galois theory, Hopf algebras, and semiabelian categories, 27–60, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004. I put these here for completeness. –  Niels Aug 24 '12 at 13:25

I think one can have the Snake Lemma in the category of (non abelian) groups by the following way.

First, I remind what is an exact sequence of pointed sets (see, for example, "Local fields" of Serre in the chapter about non abelian cohomology) : a pointed set is just a set $A$ with a based point $x_A$. A morphism of pointed sets is defined to be a map sending the based point to the based point. A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is said to be exact if $f(A) = g^{-1}(x_C)$.

Now let's go back to the snake Lemma. A cokernel is a pointed set with the class of $0$ as based point (I denote $Coker(f:A \rightarrow B)$ to be the left cosets $B/Im(f)$ but it also works for right cosets, and I call it cokernel even if it is NOT the cokernel in the theory of category). One can check that the Snake Lemma holds in the sense I have given above (with the same proof as Alex said).

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