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.

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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 ∈ S. (2) The action is said to be doubly transitive if |S| ≥ 2 ...

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