The question in the name of Normal Subgroup and Isomorphism
 
  
 
  
 
 
 
 
 
 
    
        
        asked 5 years ago  
 
 hayyambey gravatar image  
 
 
 
 
    
        
        updated 4 years ago  
 
 FrédéricC gravatar image  
 
 
 
 
 
Let A and B be two groups.
Show that set N = { (a,1):a∈A }  is the normal subgroup of A x B and 

that the A x B /N quotient group is isomorph to B. 
 
if you help me, i'll be exulted. 
 
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Homework ?
Emmanuel Charpentier gravatar imageEmmanuel Charpentier (
    
        5 years ago
    )
yes i can observe that
N=ker(π)
where π:A×B→B is the Projection map of the second factor B.
The kernel of an homomorphism of groups is always normal, so N is a normal subgroup.
Moreover, by the First isomorphism theorem, it holds
(A×B)/ker(π)≅B ... but is it enough?
hayyambey gravatar imagehayyambey (
    
        5 years ago
    )
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