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Do people have good examples where it's "easy" to compute the group-completion of a commutative monoid, but for which the monoid itself is still rather mysterious? This happens all the time in K- …

If one considers the distinguishing feature of the sign homomorphism $S_n \to \mathbb{Z}/2$ to be that it is the canonical map from $S_n$ to its abelianization, then there is nothing analogous for $S_ …

Just to make this a little more visible: for a proof that every autoequivalence of the category of groups is naturally isomorphic to the identity, see page 31 of Peter Freyd's book "Abelian Categories …

The wikipedia page Covering groups of the alternating and symmetric groups gives explicit presentations for the double covers of the symmetric group Sn (n ≥ 4). Can someone provide a similar presenta …

I think there is something to this idea that abelian groups are only groups "by accident". Here is at least a construction of the category of commutative monoids which does not mention the monoidal o …

Your functor –op is naturally isomorphic to the identity, via the natural transformation G → Gop sending g to g-1. (If you asked about monoids, not groups, then –op would be a nontrivial autoequivale …

It might be helpful to note that, in two dimensions, a shear transformation is exactly one whose Jordan canonical form is $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ or $\begin{pmatrix}1&0\\0&1\end{pmatri …

Eric already explained exactly how the suspension of a space and in particular S1 is a cogroup object in the homotopy category of pointed spaces. I'll just point out that we know it must be one, by Y …

Your questions are related to Bohr compactification, a left adjoint to the inclusion of compact (= compact Hausdorff) groups into all topological groups. A discrete group G can be embedded into a com …

Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators. This is an instance of Lawvere's theory of "theories". See this nLab entry for a dis …

Let G be a finite group of order n. Must every automorphism of G have order less than n? (David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil d …

Z/2 and Z/2 x Z/2 have the same identical relations: those words w such that every variable in w appears an even number of times. (This is obviously sufficient for w to be an identical relation. If …