2-Generator conditions in linear groups by Wehrfritz B.A.F.

By Wehrfritz B.A.F.

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For m > 1, the groups G(m, m, n) and G(m, 1, n) can be generated by n reflections; 36 2. The groups G(m, p, n) however, the minimum number of reflections required to generate G(m, p, n), for p = 1, m is n + 1. In particular, we have G(1, 1, n) = r1 , r2 , . . , rn −1 Sym(n), G(m, m, n) = s, r1 , r2 , . . , rn −1 , G(m, 1, n) = t, r1 , r2 , . . , rn −1 , and p G(m, p, n) = s, t , r1 , r2 , . . , rn −1 for p = 1, m. 8. Invariant polynomials for G(m, p, n) In the next chapter we consider group actions on polynomials in considerable detail and so for now we restrict our attention to G(m, p, n) and describe its action on polynomials P (X1 , .

That every maximal ideal of S has this form. Furthermore, for g ∈ G we have g(mv ) = mg (v ) . Let F1 , . . e. polynomial functions on V . We wish to consider properties of the map ϕ : V → C defined by ϕ(v) = (F1 (v), . . , F (v)). There is a unique algebra homomorphism ϕ∗ : C[X1 , . . , X ] → S such that ϕ∗ (Xi ) = Fi for i = 1, . . , and for P ∈ C[X1 , . . , X ] we have ϕ∗ (P ) = P ϕ, whence ϕ∗−1 (mv ) = mϕ(v ) for all v ∈ V . It follows that ϕ∗ is injective if and only if there is no polynomial P = 0 such that P (F1 , .

Xnm ) are invariants of G(m, p, n) m /p and so is the polynomial σn := en = (X1 X2 · · · Xn )m /p . 22. The polynomials σ1 , σ2 , . . , σn are algebraically independent. m /p and the result Proof. The proof is by induction on n. If n = 1, then σ1 = X1 is clear. So suppose that n > 1 and that the result holds for all groups G(m , p , n ) with n < n. By way of contradiction, suppose that H ∈ C[Y1 , . . , Yn ] is a non-zero polynomial of smallest degree in Yn such that H(σ1 , σ2 , . . , σn ) = 0.

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