Identity in Group

Algebra Level 3

The set R × R \mathbb R \times {\mathbb R}^* along with the operation * defined as follows is a group.

( a , b ) ( c , d ) = ( a d + b c , b d ) (a,b)*(c,d) = (ad+bc, bd)

If ( p , q ) (p,q) is the identity element of this group, find ( p + q ) (p+q) .


The answer is 1.

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2 solutions

Jc 506881
Jan 21, 2018

If ( p , q ) (p, q) is the identity, then ( a , b ) ( p , q ) = ( a , b ) (a,b)*(p,q) = (a,b) . So, a q + b p = a aq+bp = a and b q = b bq = b . These two equations imply that p = 0 p = 0 and q = 1 q = 1 .

If I = ( p , q ) I = \left( p, q \right) is the identity, it must be the case that ( p , q ) ( p , q ) = ( 2 p q , q 2 ) = ( p , q ) \left( p, q \right) * \left( p, q \right) = \left( 2pq, q^{2} \right) = \left( p, q \right) . This is only solvable if q = 1 q = 1 and p = 0 p = 0 , since the set is R × R \mathbb R \times {\mathbb R}^* which prevents q q from also being zero. Therefore I = ( 0 , 1 ) I = \left( 0, 1 \right) .

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