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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 : G 1 G 2 , : G 2 G 1 be group homomorphisms so that ( g ) = g for all g G 1 . (i) Prove that is one-to-one and is onto; (ii) Show that G 2 = ( G 1 ) ker . (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: 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 , H 1 , H 2 , and H 3 such that all of the following conditions hold: (i) H H 1 H 2 (ii) H = 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 6 = 1 is a normal subgroup of G . Prove that Z ( G ) H 6 = 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 ?...
View Full Document

This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

Page1 / 16

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online