Unformatted text preview: . (iv) Let g : H → K be a monomorphism. Then for any two homomorphsims f 1 ,f 2 : G → H , the equality g ◦ f 1 = g ◦ f 2 implies that f 1 = f 2 . (4) If ( U,ε ) is a universal pair for a group G and h ∈ Aut( U ) show that ( U,hε ) is also universal for G . Conversely, if ( U,ε 1 ) is universal for G show that ε 1 = hε for some h ∈ Aut( U ). (5) Find G if G = S 3 , S 4 , or A 4 . (6) Prove the lemma from class: (i) If G ≤ H ≤ G show that H C G . (ii) Show that if K C G , then K C G . (iii) Suppose f : G → H is an epimorphism, with ker f = K . Show that H is abelian if and only if G ≤ K . (7) (a) Find the derived series for S 4 . (b) Show that S n = A n if n 6 = 2. Conclude that S n is not solvable if n ≥ 5. (8) Show that any ﬁnite pgroup is solvable. (9) If  G  = p 2 q for primes p and q , show that G is solvable....
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 Fall '08
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 Algebra, Sets, Dr. Matthew Macauley, G K., Permutation groups G1

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