Let G be a group and let a, b ? G. Prove that there exist elements x and y of G such that ax = b and ya = b. Solution Theorem. If G is a group and a and b are elements of G , then there exist elements x and y of G such that ax = b and ya = b . Proof. Let a and b be given elements of a group G . Then both x -1 and y -1 , the inverses of a and b , exist. Let x = a -1 b , and let y = ba -1 . Suppose e is the identity element of G . Then we have ax = a ( a -1 b ) = ( aa -1 ) b = eb = b , so that ax = b . Also, we have ya = ( ba -1 ) a = b ( a -1 a ) = be = b , so that ya = b . This completes the proof. .

1. Let G be a group and let a, b ? G. Prove that there exist elements x and y of G such that ax = b
and ya = b.
Solution
Theorem. If G is a group and a and b are elements of G , then there exist elements x and y of G
such that ax = b and ya = b .
Proof. Let a and b be given elements of a group G . Then both x -1
and y -1
, the inverses of a and
b , exist.
Let x = a -1
b , and let y = ba -1
.
Suppose e is the identity element of G .
Then we have
ax = a ( a -1
b ) = ( aa -1
) b = eb = b ,
so that ax = b .
Also, we have
ya = ( ba -1
) a = b ( a -1
a ) = be = b ,
so that ya = b .
This completes the proof.