理科数学作业代写 Mathematics代写

School of Mathematics and Statistics

MT4003 Groups

Problem Sheet VII: Simple Groups

理科数学作业代写 Prove that if G is a simple group, and Ø : G → H is a non-trivial homomorphism (i.e. im Ø≠ 1) then Ø is injective.

1.Show that any simple abelian group is cyclic of prime order.  理科数学作业代写

2.Let n > 5, and let N be a non-trivial normal subgroup of S_n.

(a) Show that N = An or N = S_n.

[Hint: If N  S_n, consider N ∩ A_n.]

(b) Prove that S_n is directly indecomposable.

(c) Describe all homomorphic images of  S_n.

3.(a) Let K be the subgroup of A₅ generated by its 5-cycles. Show that K is a normal subgroup of A₅ and hence deduce that K = A₅.  理科数学作业代写

(b) Suppose that H is a subgroup of S₅ such that |S₅: H| < 5. Let σ be any 5-cycle. By considering cosets of H containing powers of σ, show that σ must lie in H.

Hence prove that H = A5 or S5.

(c) Show that A5 has no proper subgroup of index less than 5.

[Hint for (b): If a coset of H contains two distinct elements from {1, σ, σ2, σ3, σ4}, show that H con-tains σ.]

4.Prove that if G is a simple group, and Ø : G → H is a non-trivial homomorphism (i.e. im Ø≠ 1) then Ø is injective.

 

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