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group - 1 Group Theory 1 1 Group Theory(84 Jan Let G be a...

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1 Group Theory 1 1 Group Theory (84 Jan.) Let G be a group, with C ( G ) its center. (1) Prove that C ( G ) is a normal subgroup of G . (2) Show that the group, I ( G ), of all inner automorphisms of G is isomorphic to G/C ( G ). (84 Jan.) If H is a p -subgroup of a finite group G , and P is any Sylow p -subgroup of G , then there exists x G so that H < xPx - 1 . In particular, any two Sylow p -subgroups of G are conjugate. (84 Jan.) Prove that the center of a nontrivial finite p -group contains more than one element. (84 Jan.) Let G 1 and G 2 be Abelian groups, and let α : G 1 G 2 , β : G 2 G 1 be group homomorphisms so that βα ( g ) = g for all g G 1 . (i) Prove that α is one-to-one and β is onto; (ii) Show that G 2 = α ( G 1 ) ker β . (84 Aug.) Let G be a finitely generated group. Prove that every proper normal subgroup of G is included in a normal subgroup of G which is maximal among all proper normal subgroups of G . (84 Aug.) i) Let G be any group. Prove that there is a normal subgroup N of G such that G/N is Abelian and N H whenever H is a normal subgroup of G such that G/H is Abelian. ii) Let G be any group and let Z ( G ) be the center of G . Prove that if G/Z ( G ) is cyclic, then G is Abelian. iii) Let p be a prime number. Prove that every finite p -group is solv- able. (84 Aug.) Describe, up to isomorphism, all the group of order 99. (84 Aug.) Let G be a finite subgroup of multiplicative group of nonzero elements of a field. Prove each of the following:
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1 Group Theory 2 i) If a, b G and the orders of a and b are relatively prime, then the order of ab is mn where m is the order of a and n is the order of b . ii) If a G and k divides the order of a , then there is c G such that the order of c is k . iii) If a , b G , then there is c G such that the order of c is divisible by both the order of a and the order of b . iv) G is cyclic. (85 Jan.) Let G be a group. Prove that G cannot have four distinct proper subgroups H 0 , H 1 , H 2 , and H 3 such that all of the following conditions hold: (i) H 0 H 1 H 2 (ii) H 0 = H 2 H 3 (iii) H 1 H 3 = G . (85 Jan.) Describe, up to isomorphism, all group of order 175. (85 Jan.) Let p be a prime number. Suppose G is a finite p -group, Z is the center of G , and N is nontrivial normal subgroup of G . Prove that N Z has more than one element. (85 Aug.) Let U be the multiplicative group of complex numbers of modulus 1. Prove that U is isomorphic to the additive group R Z . (85 Aug.) Suppose that G is a group of order p n where p is a prime and H = 1 is a normal subgroup of G . Prove that Z ( G ) H = 1, where Z ( G ) is the center of G . (86 Jan.) Let C * be the multiplicative group of nonzero complex numbers and let n 1 be an integer. How many subgroups of C * have order n ? What are they? Prove that you have found all of them. (86 Jan.) If G is a group with 30 elements, then prove that G has a normal subgroup N with 1 < | N | < 30.
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