A workbook in higher algebra by David B Surowski

By David B Surowski

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Using Exercises 16 and 18, prove that the alternating groups of degree ≥ 6 are simple. 20. Let G act on the set {1, 2, . . , n}, let F be a field and let V be the F-vector space with ordered basis (v1 , v2 , . . , vn ). As we have already seen, G acts on V via the homomorphism φ : G → GL(V ). Set V G = {v ∈ V | φ(g)v = v}. (a) Show that dim V G = the number of orbits of G on {1, 2, . . , n}. (b) Let V1 ⊆ V be a G-invariant subspace of V ; thus G acts as a group of linear transformations on the quotient space V /V1 .

Let G act transitively on the set X and assume that Y ⊆ X has the property that for all g ∈ G, either gY = Y or gY ∩ Y = ∅. Show that the distinct subsets of the form gY form a system of imprimitivity in X. 14. Let G be a group acting transitively on the set X, let x ∈ X, and let Gx be the stabilizer in G of x. , is not properly contained in any proper subgroup of G). (Hint: If {Xα } is a system of imprimitivity of G, and if x ∈ Xα , show that the subgroup M = StabG (Xα ) = {g ∈ G| gXα = Xα } is a proper subgroup of G properly containing Gx .

Thus, by hypothesis, z commutes with x and y. (e) Show that [y −1 , x] = [x, y]. ) (f) Show that (xy −1 )p = e. 7 24. Here’s an interesting simplicity criterion. Let G be a group acting primitively on the set X, and let H be the stabilizer of some element of X. Assume (i) G = G , (i) H contains a normal solvable subgroup A such that G is generated by the conjugates of A. Prove that G is simple. 25. Using the above exercise, prove that the groups PSL2 (q) q ≥ 4 are all simple groups. 26. Let G be a group and let Z = Z(G).

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