s09_mthsc851_hw04

# s09_mthsc851_hw04 - (iv Let g H → K be a monomorphism...

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 4 Due Friday Feb. 13, 2009 (1) Suppose G is a ﬁnite group, H C G , and P is a p -Sylow subgroup of H . Set N = N G ( P ). Show that G = NH . (2) Permutation groups G 1 and G 2 acting on sets S 1 and S 2 are called permutation isomorphic if there exist an isomorphism θ : G 1 G 2 and a bijection φ : S 1 S 2 such that ( θx )( φs ) = φ ( xs ) for all x G 1 and s S 1 . In other words, the following diagram commutes: S 1 x / φ ± S 1 φ ± S 2 θx / S 2 Deﬁne two group actions of a group G on itself as follows: (i) the action of x G is left multiplication by x ; (ii) the action of x G is right muliplication by x - 1 . Show that the two actions are permutation isomorphic. (3) For each of the following statements, prove or give a counterexample. (i) Let f : G H be an epimorphism. Then for any two homomorphsims g 1 ,g 2 : H K , the equality g 1 f = g 2 f implies that g 1 = g 2 . (ii) Let f : G H be a monomorphism. Then for any two homomorphsims g 1 ,g 2 : H K , the equality g 1 f = g 2 f implies that g 1 = g 2 . (iii) Let g : H K be an epimorphism. 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
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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 p-group is solvable. (9) If | G | = p 2 q for primes p and q , show that G is solvable....
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