Assignment-1 (MTH-610) Group Theory Normal subgroups, Group homomorphisms (1) Show that if G is a group of even order then there are exactly an odd number of elements of order 2. (2) Let p be a prime number. Prove Wilson’s theorem, (p − 1)! ≡ −1(mod p) (3) Let p be a fixed prime number and υ : Q× → Z be a map defined as m υ( ) = r n m where r is the maximum power of p appearing in m n i.e. to say n = 0 0 0 pr m n0 , (p, m ) = 1, (p, n ) = 1. Is υ a group homomorphism? If we modify the above map to υ :Q→Z∪∞ by defining υ(0) = ∞ then will it be a homomorphism ? (4) Let G be a group of order 12. Show that either G has a normal subgroup of order 3 or G is isomorphic to A4 . Solvable groups, Jordan Hölder theorem (1) Find a composition series for A4 . Deduce that it is solvable. (2) Show that a group G is solvable if and only if G(n) is the trivial group for some n ≥ 0. Note that G(1) = [G, G], G(i) = [G(i−1) , G(i−1) ], i > 1. (3) Show that The Jordan-Hölder theorem implies the fundamental theorem of arithmetic. (4) Show that the group of all upper triangular matrices of the form 1 a b 0 1 c 0 0 1 where a, b, c ∈ R is solvable. Free Groups/Fintely presented groups (1) Describe the universal property of free groups. (2) Show that SL2 (Z) is a generated by the following two matrices, 0 −1 1 1 1 0 0 1 Is SL2 (Z) a free group on these generators? (3) Prove that every group has a presentation. (4) Find the generators and relations for GL2 (F2 ). 1 2 Group Actions (1) (2) (3) (4) (5) Show that all cycles of the same length in Sn are conjugates. Describe the class equation for A4 . If G is a non-trivial p-group then show that it has non-trivial center. Give examples of faithful, transitive and free group actions. Let H and K be two subgroups of G. Then show that the action of G by left multiplication on the coset spaces G/H and G/K are equivalent if H and K are conjugate subgroups. Sylow’s theorem Let p be a prime number. (1) Classify all groups of order 15. (2) Let G := GLn (Fp ). What is the order of a Sylow-p subgroup of G? How many Sylow p subgroups are in G? Give one example of a Slow p subgroup in G. (3) Show that any group of order 45 is abelian.