Indian Institute of Science Education and Research Pune Assignment II
Algebra I
Exercise 1. Let G acts on a set S. The action is said to be transitive if there is only one orbit. (1) Let H be a subgroup of G and H acts on S transitively. Then G = H.StabG (a) for any a ∈ S. (2) The action is said to be doubly transitive if |S| ≥ 2 and for any pair (a, b) and (c, d) in S × S \ 4 there exists g ∈ G such that ga = c and gb = d. Show that G is doubly transitive if and ony if it is transitive and StabG (a) is transitive on S \ {a} for all a ∈ S. (3) Suppose G is transitive on S. A proper subset B ⊂ S with |B| ≥ 2 is said to be a block if gB = B or gB ∩ B = ∅ for all g ∈ G. A transitive action is said to be primitive if it has no blocks. Show that a transitive action is primitive if and only if StabG (a) is a maximal subgroup for all a ∈ G. (4) If G is doubly transitive on S then G is primitive. (5) Let G acts primitively on S and N is a normal subgroup of G. Further suppose N is not contained in the kernel of the action. Then N is transitive on S. Exercise 2 (Iwasawa Lemma). Let G acts on S, faithfully, transitively and primitively. Let [G, G] = G. Fix s ∈ S and denote H = StabG (s). Suppose there is a solvable subgroup K which is normal in H with property that G is generated by all conjugates of K. Then G is simple. Exercise 3. Compute centralizers of all elements in symmetric group Sn . Exercise 4. Classify all groups of order p2 and p3 , where p is a prime. Exercise 5. Show that the group P SL2 (Z) is (2, 3) generated, i.e, it is generated by two elements, one of order 2 and another of order 3. Exercise 6. Compute automorphism group of all groups up to order 10.
Indian Institute of Science Education and Research Pune
Indian Institute of Science Education and Research Pune. Assignment II. Algebra I. Exercise 1. Let G acts on a set S. The action is said to be transitive if there is only one orbit. (1) Let H be a subgroup of G and H acts on S transitively. Then G = H.StabG(a) for any a â S. (2) The action is said to be doubly transitive if |S| ⥠2 ...
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Indian Institute of Science Education and Research. Assignment 4. MTH 320 - Vector Space, Rings and Modules. Dimension of vector space is n. Exercise 1. Let T â End(V ). Further suppose V = V1 â V2 â···â Vk such that T(Vi) â Vi. Denote T
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