From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7347 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Two questions Date: Fri, 22 Jun 2012 23:09:21 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1340457316 2227 80.91.229.3 (23 Jun 2012 13:15:16 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 23 Jun 2012 13:15:16 +0000 (UTC) To: "categories" Original-X-From: majordomo@mlist.mta.ca Sat Jun 23 15:15:16 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SiQBF-0002q7-IQ for gsmc-categories@m.gmane.org; Sat, 23 Jun 2012 15:15:13 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50559) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SiQAh-0004gT-Bb; Sat, 23 Jun 2012 10:14:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SiQAh-0000MH-C7 for categories-list@mlist.mta.ca; Sat, 23 Jun 2012 10:14:39 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7347 Archived-At: On Fri, 22 Jun 2012 01:32:10 +0200, George Janelidze wrote, inter alia: > 6. Concerning your second question: What you saw (with a special argume= nt > for 2) is on the same Page 21 of the above-mentioned Eilenberg--Moore paper. > The argument was used to prove that every epimorphism of groups is > surjective with no mention of regular monomorphisms, but in fact they p= rove > that, for every homomorphism > = > j : H --> G, there exist two homomorphisms > = > k, l : G --> P with k(g) =3D L(g) only when g is in j(H). > = > This also appears as Exercise 5 of Section 5 of Chapter I in Mac Lane's= > "Categories for the Working Mathematician", with very precise hints. What the origins of the argument Mac Lane outlines here are, I don't know= =2E = I do seem to recall that I first saw more or less such an argument in a = graduate course Sammy gave at Columbia during the "golden era" 1958-1963,= = and that Sammy himself, probably at a Bowdoin NSF summer semester on = categories, revealed his deft trick by which to convert the argument for = the "index greater than 2" case to a uniform argument ignoring the = subgroup's index. Let me record that argument here. Given are a group G, a subgroup H of G, and an element a in G \ H. P is to be the group of permutations of the underlying set of the left = G-set got by forming the coproduct = G/H + 1 of the principal left G-set G/H of left cosets xH of H in G (x =E2=88=88 = G) = with a (trivial) terminal left G-set 1 =3D {*} (assume * =E2=88=89 G/H): P =3D perm(|G/H| =E2=88=AA {*}) =3D (|G/H| =E2=88=AA {*})! . One permutation in particular is to be singled out for attention: the = transposition t =E2=88=88 P interchanging * with the coset H itself. Let me use r: G --> P for the result of composing the regular = representation of G by left action (g, xH) |-> (gx)H on the cosets of H = with the obvious injection of (|G/H|)! into (|G/H| =E2=88=AA {*})! -- thu= s: [r(g)](xH) =3D (gx)H , [r(g)](*) =3D * . And let me write s: G --> P for the result of conjugating by t the variou= s = assorted values of r -- thus: [s(g)](xH) =3D t([r(g)](t(xH))) , [s(g)](*) =3D t([r(g)](t(*))) =3D t([r(g)](H)) =3D t(gH) . The end-game strategy is now this: (i) for h =E2=88=88 H: r(h) =3D s(h) ; yet (ii) for g =3D a, [r(a)](*) =3D * but [s(a)](*) =3D aH (whence r =E2=89=A0= s). Details: Bear in mind that if, for x in G and h in H, we have (hx)H =3D H= , = we must have hx in H, whence also x in H so that xH =3D H. Then for (i): (a) if xH =E2=89=A0 H -- [s(h)](xH) =3D t([r(h)](xH)) =3D t((hx)H) =3D (h= x)H =3D [r(h)](xH) ; (b) if xH =3D H -- [s(h)](H) =3D t([r(h)](*)) =3D t(*) =3D H =3D [r(h)](H= ) ; (c) and at * -- [s(h)](*) =3D t([r(h)](H)) =3D t(H) =3D * =3D [r(h)](*) .= And bear in mind also that aH =E2=89=A0 H because a =E2=88=89 H; thus: for (ii) -- [s(a)](*) =3D t([r(a)](H)) =3D t(aH) =3D aH =E2=89=A0 * =3D [= r(a)](*) . By the way, if the subgroup H had index 3 or more in G, one need not = require recourse to any external element * as above -- one could let any = coset other than H and aH (when there are such) play the role of *, and, = with but a few additional wrinkles (unless I am mistaken (which is always= = possible :-) )), I believe that is essentially how the proof George cites= = from Mac Lane's exercise works. Cheers, -- Fred [NB: I'm writing =E2=88=88 for an element symbol, =E2=88=89 for a crossed= -out = element symbol, =E2=88=AA for a union symbol, and =E2=89=A0 for a crossed= -out equal = sign. With luck these HTML glyph constructs will simply display as the = glyphs they're meant to represent; and if not, they're no more painful to= = decipher than their TeX counterparts, which HTML can't display as glyphs.= ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]