With regard to the second question, the problem seems to be for epic. I found the solution somewhere and put it as exercise 8 of section 6.1 of my Topology and Groupoids book (all editions). Here is the outline argument (I have not checked it lately!): Prove that in the category $\grp$ of groups, a morphism $f : G \to H$ is monic if and only if it is injective; less trivially, $f$ is epic if and only if $f$ is surjective. [Suppose $f$ is not surjective and let $K = \Im f$. If the set of cosets $H/K$ has two elements, then $K$ is normal in $H$ and it is easy to prove $f$ is not epic. Otherwise there is a permutation $\gamma$ of $H/K$ whose only fixed point is $K$. Let $\pi : H \to H/K$ be the projection and choose a function $\theta : H/K \to H$ such that $\pi \theta = 1$. Let $\tau : H \to K$ be such that $x = (\tau x)(\theta\pi x)$ for all $x$ in $H$ and define $\lambda : H \to H$ by $x \mapsto (\tau x) (\theta\gamma\pi x)$. The morphisms $\alpha, \beta$ of $H$ into the group $P$ of all permutations of $H$, defined by $\alpha(h)(x) = hx$, $\beta(h) = \lambda^{-1}\alpha(h)\lambda$ satisfy $\alpha h = \beta h$ if and only if $h \in K$. Hence $\alpha f = \beta f$]. Ronnie On 21/06/2012 14:34, Michael Barr wrote:
Googling around, I have come on several claims that there are no non-trivial injectives in the category of groups (e.g., Mac Lane in the 1950 Duality for groups paper credits Baer with an elegant proof, but gives no hint of what it might be and Baer's earlier paper on injectives doesn't mention it). I have not come on any proof of this, however.
Somewhere I have seen a proof that all monics in the category of groups are regular. I think it was in a paper by Eilenberg and ??? and it needed a special argument if there were elements of order 2. Can someone help me find this?
Michael
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