a galois related problem
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From: Steve Dalton on 12 Feb 2010 23:28 [...] > That seems reasonable to me. > Note that I was not arguing that the assertion > about polynomials with galois group D_7 was > incorrect; > I was merely suggesting that the argument was more > complicated > than that given. > > Actually, I would have thought that if p is an odd > prime > one could see more easily that the natural embedding > of D_p in S_p > is the only one (up to conjugacy). > One can take the p-cycle in D_p to be s=(1,2,...,p). > If t is an element of order 2 in D_p then > tst^{-1} is in <s>, ie tst^{-1} = s^r for some r. > > But then t^2st^{-2} = s^{r^2} = s => r^2 = 1 mod p => > r = +/-1. > Thus r = -1 (since D_p is not abelian), > and it follows that one has the natural embedding. > > I think this argument also applies if n = 2p, with p > an odd prime. > But in all other cases there are solutions of r^2 = 1 > mod n > other than +/-1, and so I think there are > non-standard > embeddings of D_n in S_n. > > > > > > -- > Timothy Murphy > e-mail: gayleard /at/ eircom.net > tel: +353-86-2336090, +353-1-2842366 > s-mail: School of Mathematics, Trinity College, > Dublin 2, Ireland Well yes, it is always true that tst^{-1} = s^{-1}, that is by definition of the dihedral group. The question is if the action on the n points is the dihedral action (treating them as vertices of a polygon, etc.). For example, in S_6 (a 2p case), the subgroup < (12)(345), (45) > is isomorphic to D_6, but clearly doesn't give a dihedral action on the 6 points (it fixes "6"). If you can assume that in the embedding of D_n in S_n, an n-cycle is included, this simplifies things a lot. In the odd n case, you get up to conjugacy the usual embedding. In the n even case, you get an action where the reflection is either in a line through two vertices (the usual dihedral action) or a line through two parallel sides (I don't think that's allowed). Steve
From: Chip Eastham on 13 Feb 2010 10:17 On Feb 13, 9:28 am, Steve Dalton <sdal...(a)math.sunysb.edu> wrote: > [...] > > > > > > > That seems reasonable to me. > > Note that I was not arguing that the assertion > > about polynomials with galois group D_7 was > > incorrect; > > I was merely suggesting that the argument was more > > complicated > > than that given. > > > Actually, I would have thought that if p is an odd > > prime > > one could see more easily that the natural embedding > > of D_p in S_p > > is the only one (up to conjugacy). > > One can take the p-cycle in D_p to be s=(1,2,...,p). > > If t is an element of order 2 in D_p then > > tst^{-1} is in <s>, ie tst^{-1} = s^r for some r. > > > But then t^2st^{-2} = s^{r^2} = s => r^2 = 1 mod p => > > r = +/-1. > > Thus r = -1 (since D_p is not abelian), > > and it follows that one has the natural embedding. > > > I think this argument also applies if n = 2p, with p > > an odd prime. > > But in all other cases there are solutions of r^2 = 1 > > mod n > > other than +/-1, and so I think there are > > non-standard > > embeddings of D_n in S_n. > > > -- > > Timothy Murphy > > e-mail: gayleard /at/ eircom.net > > tel: +353-86-2336090, +353-1-2842366 > > s-mail: School of Mathematics, Trinity College, > > Dublin 2, Ireland > > Well yes, it is always true that tst^{-1} = s^{-1}, that is by definition of the dihedral group. The question is if the action on the n points is the dihedral action (treating them as vertices of a polygon, etc.). For example, in S_6 (a 2p case), the subgroup > < (12)(345), (45) > is isomorphic to D_6, but clearly doesn't give a dihedral action on the 6 points (it fixes "6"). > > If you can assume that in the embedding of D_n in S_n, an n-cycle is included, this simplifies things a lot. In the odd n case, you get up to conjugacy the usual embedding. In the n even case, you get an action where the reflection is either in a line through two vertices (the usual dihedral action) or a line through two parallel sides (I don't think that's allowed). > > Steve Timothy's question was a good one, which impelled me to slap out a less than polished reply! Note that the Galois group acts transitively on the roots of an irreducible polynomial, so that examples like the "bad" embedding of D_6 in S_6 Steve describes above won't arise in the context of Galois theory of splitting fields. [Element "6" is in an orbit unto itself.] regards, chip
From: Steve Dalton on 13 Feb 2010 00:25 > [...] > > > That seems reasonable to me. > > Note that I was not arguing that the assertion > > about polynomials with galois group D_7 was > > incorrect; > > I was merely suggesting that the argument was more > > complicated > > than that given. > > > > Actually, I would have thought that if p is an odd > > prime > > one could see more easily that the natural > embedding > > of D_p in S_p > > is the only one (up to conjugacy). > > One can take the p-cycle in D_p to be > s=(1,2,...,p). > > If t is an element of order 2 in D_p then > > tst^{-1} is in <s>, ie tst^{-1} = s^r for some r. > > > > But then t^2st^{-2} = s^{r^2} = s => r^2 = 1 mod p > => > > r = +/-1. > > Thus r = -1 (since D_p is not abelian), > > and it follows that one has the natural embedding. > > > > I think this argument also applies if n = 2p, with > p > > an odd prime. > > But in all other cases there are solutions of r^2 = > 1 > > mod n > > other than +/-1, and so I think there are > > non-standard > > embeddings of D_n in S_n. > > > > > > > > > > > > -- > > Timothy Murphy > > e-mail: gayleard /at/ eircom.net > > tel: +353-86-2336090, +353-1-2842366 > > s-mail: School of Mathematics, Trinity College, > > Dublin 2, Ireland > > Well yes, it is always true that tst^{-1} = s^{-1}, > that is by definition of the dihedral group. The > question is if the action on the n points is the > dihedral action (treating them as vertices of a > polygon, etc.). For example, in S_6 (a 2p case), the > subgroup > < (12)(345), (45) > is isomorphic to D_6, but clearly > doesn't give a dihedral action on the 6 points (it > fixes "6"). > > If you can assume that in the embedding of D_n in > S_n, an n-cycle is included, this simplifies things a > lot. In the odd n case, you get up to conjugacy the > usual embedding. In the n even case, you get an > action where the reflection is either in a line > through two vertices (the usual dihedral action) or a > line through two parallel sides (I don't think that's > allowed). > > Steve Nevermind, of course it's allowed; and thus we can say, for all n, if a particular embedding of D_n into S_n has an n-cycle in its image, it is, up to conjugacy, the "usual" embedding (so it has a dihedral action on the n points). This is of course always the case when n is prime. I guess another way of seeing this is to realize the normalizer of an n-cycle in S_n is the holomorph, and there is precisely one coset consisting of elements which invert the n-cycle. The union of the subgroup generated by the n-cycle, and this coset, is the "usual" dihedral group. Steve
From: victor_meldrew_666 on 13 Feb 2010 11:57 On 13 Feb, 12:15, Timothy Murphy <gayle...(a)eircom.net> wrote: > Steve Dalton wrote: > > A simpler argument for (odd) prime n: > > > D_n in S_n is a subgroup generated by an n-cycle and a non-centralizing, > > normalizing involution. Such an involution can fix at most one point. > > What exactly do you mean by "D_n in S_n"? > There is a natural way of embedding D_n in S_n, > but it is not obvious that this is the way the Galois group in question > is embedded in S_n. When n is composite, there may be other ways of embedding D_n in S_n. For example it's possible to embed D_15 in S_8. However for this Galois-theoretical application, one needs an emebedding of D_n as a *transitive* subgroup of S_n, and I think that this can only be done in the "standard" way.
From: Jim Heckman on 14 Feb 2010 03:35
On 13-Feb-2010, "victor_meldrew_666(a)yahoo.co.uk" <victor_meldrew_666(a)yahoo.co.uk> wrote in message <a51ed533-4c32-477c-ab05-46485414e94f(a)z19g2000yqk.googlegroups.com>: > On 13 Feb, 12:15, Timothy Murphy <gayle...(a)eircom.net> wrote: > > > Steve Dalton wrote: > > > > A simpler argument for (odd) prime n: > > > > > D_n in S_n is a subgroup generated by an n-cycle and a > > > non-centralizing, > > > normalizing involution. Such an involution can fix at most one point. > > > > What exactly do you mean by "D_n in S_n"? > > There is a natural way of embedding D_n in S_n, > > but it is not obvious that this is the way the Galois group in question > > is embedded in S_n. > > When n is composite, there may be other cays of embedding D_n > in S_n. For example it's possible to embed D_15 in S_8. Not to mention D_6 in S_6, as others have pointed out. > However > for this Galois-theoretical application, one needs an emebedding > of D_n as a *transitive* subgroup of S_n, and I think that > this can only be done in the "standard" way. Right. When n is odd there's only one (up to equivalency) such embedding, corresponding to the single conjugacy class of involutions in D_n, which of course are the point stabilizers of the action. When n is even there are two inequivalent embeddings, corresponding to the two conjugacy classes of non-central involutions, but the images of the two embeddings are permutation isomorphic, i.e., the "same" subgroup of S_n (up to conjugacy). -- Jim Heckman |