The set R × R ∗ along with the operation ∗ defined as follows is a group.
( a , b ) ∗ ( c , d ) = ( a d + b c , b d )
If ( p , q ) is the identity element of this group, find ( p + q ) .
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If I = ( p , q ) is the identity, it must be the case that ( p , q ) ∗ ( p , q ) = ( 2 p q , q 2 ) = ( p , q ) . This is only solvable if q = 1 and p = 0 , since the set is R × R ∗ which prevents q from also being zero. Therefore I = ( 0 , 1 ) .
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If ( p , q ) is the identity, then ( a , b ) ∗ ( p , q ) = ( a , b ) . So, a q + b p = a and b q = b . These two equations imply that p = 0 and q = 1 .