a galois related problem
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From: groupoide on 19 Feb 2010 09:53 > If S_4 (or any finite group G) occurs as a Galois group over Q at all > (and S_4 does of course), then any transitive permutation > representation of G occurs as a Galois group over Q, so your two > examples must both arise as Galois groups. Now I suppose that G is a group Galois group over Q. suppose K is a Galois extension of Q with |K:Q|=|G| and Galois group G Let a one primitif element of K extension of Q. Let P the minimal polynome of a on Q. Let E the set of roots of P. Can you explain the definition of transitive permutation représentation of G ? Is it an application from G to Sym(G)? Is it an application from G to Sym(E)? > To see this, suppose K is a Galois extension of Q with |K:Q|=|G| and > Galois group G. The transitive permutations of G are equivalent to the > coset actions on subgroups of G and are in 1-1 correspondence with the > conjugacy classes of subgroups of G> Derek Holt.- Can you explain "transitive permutations of G" ? "transitive permutations of G" is it a transitive subgroup of Sym(E)? Can you explain "coset actions on subgroups of G" et what is the equivalence between "transitive permutations of G" and "coset actions on subgroups of G" Many thanks.
From: I.M. Soloveichik on 19 Feb 2010 00:24 > Can anybody give an example of equation of degree n > with a Galois group of order 2n but not Dn? Here is an example with 3 real roots of degree 9 x^9-4*x^3+3*x^6+1 and galois group of order 18 but not the dihedral group, generated by (1 2)(4 5)(7 8), (1 2 9)(3 4 5)(6 7 8), (3 6 9)(1 4 7)(2 5 8) im
From: Derek Holt on 19 Feb 2010 11:17
On 19 Feb, 14:53, groupoide <soumil...(a)gmail.com> wrote: > > If S_4 (or any finite group G) occurs as a Galois group over Q at all > > (and S_4 does of course), then any transitive permutation > > representation of G occurs as a Galois group over Q, so your two > > examples must both arise as Galois groups. > > Now I suppose that G is a group Galois group over Q. > suppose K is a Galois extension of Q with |K:Q|=|G| and Galois group G > Let a one primitif element of K extension of Q. > Let P the minimal polynome of a on Q. > Let E the set of roots of P. > > Can you explain the definition of transitive permutation > représentation of G ? A subgroup G of the symmetric group Sym(X) acting on the set X is transitive if there is a unique orbit. That is, if: for all i,j in X, there exists g in G with g(i)=j. The Galois group of an irreducible polynomial considered as a subgroup of the symmetric group on the set of roots of the polynomial is transitive. A permutation representation (or action) of a group G is a homomorphism from G to Sym(X) for some set X. It is transitive if its image is transitive. > > Is it an application from G to Sym(G)? > Is it an application from G to Sym(E)? > > > To see this, suppose K is a Galois extension of Q with |K:Q|=|G| and > > Galois group G. The transitive permutations of G are equivalent to the > > coset actions on subgroups of G and are in 1-1 correspondence with the > > conjugacy classes of subgroups of G> Derek Holt.- > > Can you explain "transitive permutations of G" ? I am sorry, that was a typo. I meant to write "The transitive permutation representations". > "transitive permutations of G" is it a transitive subgroup of Sym(E)? > > Can you explain "coset actions on subgroups of G" et what is the > equivalence between "transitive permutations of G" and "coset actions > on subgroups of G" Let H be a subgroup of G, and let X be the set of left cosets xH of H in G. Then there is a transitive permutation representation phi:G- >Sym(X) for which phi(g) is the permutation that maps each coset xH to gxH. (So the action is by left multiplication.) This type of representation is called a coset action. Two permutation representations phi:G->Sym(X) and psi:G->Sym(Y) of the same group G are said to be equivalent if the there is a bijection tau:X->Y such that psi(g)( tau(x) ) = tau( phi(g)(x) ) for all g in G, x in X (hope I've got that right!) There is a theorem that says that every transitive permutation representation of G is equivalent to a coset action. Also, the coset actions on the cosets of the subgroups H1, H2 of G are equivalent if and only if H1 and H2 are conjugate in G. Derek Holt. |