a galois related problem
in [Math]
|
Prev: Boundness of solutions of ODE systems
Next: formalizing what math assumed as the definition of "finite-number"#424 Correcting Math
From: Kent Holing on 13 Feb 2010 23:12 A follow-up question: Of all transitive subgroups of Sn and of order 2n; does it exist one such not isomorphic to Dn?
From: Timothy Murphy on 14 Feb 2010 10:20 Kent Holing wrote: > A follow-up question: > Of all transitive subgroups of Sn and of order 2n; does it exist one such > not isomorphic to Dn? What about Q_8 (group of quaternions +/-{1,i,j,k}) acting on Q_8/{+/-1} ? -- 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
From: Steve Dalton on 14 Feb 2010 00:23 > A follow-up question: > Of all transitive subgroups of Sn and of order 2n; > does it exist one such not isomorphic to Dn? Sure, in S_8, the subgroup <(12345678), (24)(37)(68)> is not dihedral. Steve
From: Chip Eastham on 14 Feb 2010 12:48 On Feb 14, 9:12 am, Kent Holing <K...(a)statoil.com> wrote: > A follow-up question: > Of all transitive subgroups of Sn and of order 2n; does it exist one such not isomorphic to Dn? If n is an odd prime, the only groups of order 2n are cyclic or dihedral (can be deduced from Sylow theory). Also if n is prime, Sym(n) does not contain a cyclic group of order 2n (by disjoint cycle representation the order of a permutation in Sym(n) is either n or coprime to n). Thus for odd prime n we know there are no subgroups of Sym(n) of order 2n not isomorphic to Dih(n), even without invoking transitivity. [But a posteriori these are transitive by virtue of having a cycle of length n. Cf. Beachy/Blair, AAOL Lemma 8.6.4] http://www.math.niu.edu/~beachy/aaol/galois.html#compute Also: For n = 4 the only subgroups of Sym(4) of order 8 are dihedral (again without invoking transitivity) and these are the three Sylow 2-subgroups of Sym(4), and all of them are transitive. For n = 6 the only subgroups of Sym(6) of order 12 are dihedral (but not all of them are transitive). regards, chip
From: Timothy Murphy on 14 Feb 2010 13:07
Timothy Murphy wrote: >> A follow-up question: >> Of all transitive subgroups of Sn and of order 2n; does it exist one such >> not isomorphic to Dn? > > What about Q_8 (group of quaternions +/-{1,i,j,k}) acting on Q_8/{+/-1} ? Sorry, that was a silly remark. I was forgetting you need a faithful and transitive action ... -- 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 |