s09_mthsc851_hw05

s09_mthsc851_hw05 - G 1 G 2 G n solvable? (7) If f : A B is...

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 5 Due Monday Feb. 23, 2009 (1) (a) Let | G | be finite and N C G . If xN G/N has order a power of p , show that there exists y G such that | y | is a power of p and yN = xN . (b) If G/N is abelian and P is a p -Sylow subgroup of G , prove that PN/N is the unique p -Sylow subgroup of G/N . (2) Let H i C G i . Prove or give a counterexample. (a) G 1 = G 2 , H 1 = H 2 G 1 /H 1 = G 2 /H 2 . (b) G 1 = G 2 , G 1 /H 1 = G 2 /H 2 H 1 = H 2 . (c) H 1 = H 2 , G 1 /H 1 = G 2 /H 2 G 1 = G 2 . (3) Give an example of groups H i , K j such that H 1 × H 2 = K 1 × K 2 and no H i is isomorphic to any K j . (4) Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of two of its proper subgroups. (5) If G is the internal direct product of subgroups G 1 and G 2 , show that G/G 1 = G 2 and G/G 2 = G 1 . (6) (a) Show that Z ( Q α G α ) = Q α Z ( G α ). (b) Show that ( G 1 × G 2 × ··· × G n ) 0 = G 0 1 × G 0 2 × ··· × G 0 n . (c) Under what circumstances is
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Unformatted text preview: G 1 G 2 G n solvable? (7) If f : A B is an equivalence in a category C and g : B A is a morphism such that gf = 1 A and fg = 1 B , show that g is unique. (8) (a) Prove that any two universal (initial) objects in a category C are equivalent. (b) Prove that any two couniversal (terminal) objects in a category C are equivalent. (9) In the category of abelian groups, show that the group A 1 A 2 together with the homo-morphisms 1 : A 1 A 1 A 2 , 1 ( x ) = ( x,e ) 2 : A 2 A 1 A 2 , 2 ( x ) = ( e,x ) is a coproduct for { A 1 ,A 2 } . Why is this not a coproduct in the category of groups?...
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