Thread: group theory abstract algebra

consider all the rationals except 1 with the binary operation a*b=a+b-ab for any a,b elements in Q\ {1}.
show that this is a group.
why is 1 excluded

Verify that satisfies all the group axioms. (Do you know what the group axioms are?) Hint for identity:

This group is isomorphic to (the group of nonzero rationals under ordinary multiplication) via the mapping for

Originally Posted by bettywhit

consider all the rationals except 1 with the binary operation a*b=a+b-ab for any a,b elements in Q\ {1}.
show that this is a group.
why is 1 excluded

thanks i figured out how to show it is a group but i am not sure why 1 is excluded

Originally Posted by bettywhit

thanks i figured out how to show it is a group but i am not sure why 1 is excluded

The group's unit is 0, since . If 1 were an element of this group, then you'd get and you have ANOTHER unit, which cannot be in a group.

Tonio

Something is wrong here.

You just cannot have two identities, even in a semigroup! Including would make a semigroup rather than a group (the element will not have an inverse). If a semigroup has an identity, then that identity is always unique.

If 1 was included, the inverse would also be undefined,