homework7 - Math 113 Section 5 Fall 2010 Homework #7 Due:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 113 Section 5 Homework #7 Fall 2010 Due: Monday, October 25 1. Section 4.3, Exercise 5: Let G be a group. If the center of G is of index n , prove that every conjugacy class has at most n elements. 2. Section 5.1, Exercise 1: Let G 1 ,G 2 ,...,G n be groups. Show that the center of a direct product is the direct product of the centers: Z ( G 1 × G 2 × ··· × G n ) = Z ( G 1 ) × Z ( G 2 ) × ··· × Z ( G n ) . Deduce that a direct product of groups is abelian if and only if each of the factors is abelian. 3. Section 5.1, Exercise 2: Let G 1 ,G 2 ,...,G n be groups and let G = G 1 × ··· × G n . Let I be a proper, nonempty subset of { 1 ,...,n } and ket J = { 1 ,...,n }- I . Define G I to be the set of elements of G that have the identity of G j in position j for all j J . (a) Prove that G I is isomorphic to the direct product of the groups G i ,i I . (b) Prove that G I is a normal subgroup of G and G/G I is isomorphic to G J . (c) Prove that
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/21/2012 for the course MATH 113 taught by Professor Ogus during the Fall '08 term at University of California, Berkeley.

Page1 / 2

homework7 - Math 113 Section 5 Fall 2010 Homework #7 Due:...

This preview shows document pages 1 - 2. Sign up to view the full document.

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