Views
3 years ago

# On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

## Lemma 3.1 Let S be a

Lemma 3.1 Let S be a subgroup of S n , n ≥ 3 that is generated by transpositions.ThenE(G, n) Sdef= 〈⃗g σ ⃗g −1 |⃗g ∈ G n , σ ∈ S〉= 〈⃗g σ ⃗g −1 |⃗g ∈ K(G, n), σ ∈ S〉K(G, n) Sdef= 〈[⃗g, σ] |⃗g ∈ G n , σ ∈ S〉= 〈[⃗g, σ] |⃗g ∈ K(G, n), σ ∈ S〉.Moreover, it is enough that σ runs through a system of generating transpositionsof S in the expressions above.PROOF. We will first assume that S = 〈τ〉 for the transposition τ = (1 2) of S n .For (g 1 , ..., g n ) ∈ G n we calculate(g 1 , ..., g n )τ(g 1 , ..., g n ) −1 = (g 1 g 2 −1 , g 2 g 1 −1 , 1, ..., 1)τ.In this case the subgroup K(G, n) S of G n is generated by (g, g −1 , 1, ..., 1), g ∈ G.Since we assumed n ≥ 3 we may consider the element (g, 1, g −1 , 1, ...). Byapplying the previous calculation to the transposition (1 3) this is also an elementof K(G, n). From(g, 1, g −1 , 1, ..., 1)τ(g, 1, g −1 , 1, ..., 1) −1 = (g, g −1 , 1, ..., 1)τwe deduce that 〈⃗gσ⃗g −1 |⃗g ∈ K(G, n), σ ∈ S〉 is generated by the same elementsas E(G, n) S . So both subgroups are equal. A similar calculation yields the resultfor K(G, n) S .We now let S be a subgroup of S n generated by transpositions. Then we canwrite σ ∈ S as a product τ 1 · ... · τ d of transpositions all lying in S. For ⃗g ∈ G nwe get( d∏ d∏⃗gσ⃗g −1 = ⃗g τ i)⃗g −1 = ⃗gτ i ⃗g −1 .i=1j=iWe have seen above that all ⃗gτ i ⃗g −1 can be written as products of of conjugates ofτ i under K(G, n). So ⃗gσ⃗g −1 can be written as a product of K(G, n)-conjugates ofelements of S.To prove the remaining assertion we assume that σ ∈ S can be written as aproduct of d transpositions of S. The case d = 1 was already done above. We canfind a transposition τ and an element ν that can be written as a product of strictlyless that d transpositions such that σ = τ · ν. Then also τντ −1 can be written as aproduct of strictly less than d transpositions and writing[⃗g, τν] = [⃗g, τ] · τ[⃗g, ν]τ −1 = [⃗g, τ] · [τ⃗gτ −1 , τντ −1 ]we can apply induction.□12

Remark 3.2 The assumption n ≥ 3 is crucial:1. If n = 1 then E(G, 1) = K(G, 1) = 1.2. If n = 2 then K(G, 2) is the subgroup of G 2 generated by (g, g −1 ) and〈⃗gσ⃗g −1 |⃗g ∈ G 2 , σ ∈ S 2 〉 = 〈(g, g −1 ), g ∈ G〉 = K(G, 2)〈⃗gσ⃗g −1 |⃗g ∈ K(G, 2), σ ∈ S 2 〉 = 〈(g 2 , g −2 ), g ∈ G〉 ≤ K(G, 2).So in this case it depends on the structure of G whether these two subgroupscoincide.3.2 PropertiesThere exists a quite different description of K(−, n) given by the followingProposition 3.3 Let n ≥ 2 be a natural number and G be an arbitrary group.ThenK(G, n) = ker ( G n → G ab )(g 1 , ..., g n ) ↦→ g 1 · ... · g nas subgroups of G n .PROOF. Lemma 3.1 tells us that K(G, n) is generated by elements of the form(1, ..., 1, g, 1, ..., 1, g −1 , 1, ..., 1). Since these elements lie in the kernel of the mapG n → G ab it follows that we already have K(G, n) ≤ ker(G n → G ab ).Conversely, suppose that (g 1 , ..., g n ) lies in the kernel of G n → G ab . Multiplyingby (1, ..., g n , g −1 n ) we obtain an element of the form (g 1 , ..., g n−1, ′ 1). Wemultiply this element by (1, ..., 1, g n−1 ′ , g′ n−1 −1 , 1). Proceeding inductively, we seethat every element of ker(G n → G ab ) can be changed by elements from K(G, n)to an element of the form (g 1, ′ 1, ..., 1). Then necessarily g 1 ′ ∈ [G, G]. This meansthat g 1 ′ is a product of commutators. Since n ≥ 2 we can write a commutator as aproduct of elements of K(G, n):[(h 1 , 1, ..., 1), (h 2 , 1, ..., 1)]= (h 1 , h 1 −1 , 1, ..., 1) (h 2 , h 2 −1 , 1, ..., 1) ((h 2 h 1 ) −1 , (h 2 h 1 ), 1, ..., 1).For computations later on we remark that for n ≥ 3 such a commutator is even acommutator of elements of K(G, n):[(h 1 , 1, ..., 1), (h 2 , 1, ..., 1)] = [(h 1 , h 1 −1 , 1, ...), (h 2 , 1, h 2 −1 , 1, ...)].Hence every element of ker(G n → G ab ) is a product of elements of K(G, n). Thisproves the converse inclusion and so we are done.□13

The Path Ahead: Fundamentals of Generative ... - Right 4 Your Type
Project Management Fundamentals
Groups of integral representation type - MSP
Generic process improvement approach applied to IT projects - PMI ...
Renewable Projects Portal for CLP Group - OSIsoft
Project Insight - Power Generation Construction Projects in Asia-Pacific
Carbon reductions generate positive ROI - Carbon Disclosure Project
Projected Costs of Generating Electricity - OECD Nuclear Energy ...
Building Next Generation Design Support Programmes - See Project
GALOIS THEORY - Tata Institute of Fundamental Research
On The Structure of Certain Galois Cohomology Groups
The Fundamental Theorem of Galois Theory and Normal Subgroups
Generic Galois groups for q-difference equations
fields and galois theory - Neil Strickland - University of Sheffield
periods of eisenstein series: the galois case - Project Euclid
Introduction to Differential Galois Theory
Galois module structure of local unit groups
Chapter 4 COVERING PROJECTIONS AND FUNDAMENTAL GROUP