s09_mthsc851_hw08

s09_mthsc851_hw08 - MTHSC 851 (Abstract Algebra) Dr....

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 8 Due Thursday April 2nd, 2009 (1) Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements is a homomorphic image of N (i.e., free objects do not always exist in the category C of nilpotent groups). (2) Let G be a group with X G , and let A be the normal subgroup generated by X , i.e., A = { N C G : X N } . Let Y = { gxg- 1 | x X, g G } . Show that A = h Y i . (3) Show that the Klein 4-group V has presentation h a,b | a 2 = b 2 = ( ab ) 2 = 1 i . (4) Show that the quaternion group Q 2 = { 1 , i, j, k } has presentation h a,b | a 4 = 1 , a 2 = b 2 , ab = ba 3 i . (5) (a) Determine the group with presentation h a,b | a 2 = 1 , b 3 = 1 , ab = ba i . (b) If G and H each have more than one element, show that G * H is an infinite group with center h e i ....
View Full Document

Ask a homework question - tutors are online