This preview shows page 1. Sign up to view the full content.
Unformatted text preview: f (( h , 1 )) , and ker f = { ( h , k ) H K : f (( h , k )) = 1 } = { ( 1 , k ) H K : k K } = K . The f rst isomorphism theorem now gives K C H K and ( H K )/ K = H . (Of course, the function H K K , sending ( h , k ) 7 k , is a homomorphism with kernel H , and this implies that H C H K . 2.98 If G is a group and G / Z ( G ) is cyclic, where Z ( G ) denotes the center of G , prove that G is abelian; that is, G = Z ( G ) . Conclude that if G is not abelian, then G / Z ( G ) is never cyclic....
View
Full
Document
This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

Click to edit the document details