Re: Two questions

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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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