Thread: Group Structure

Show that a*b=a+b+ab defines a group structure on G=R\{-1} using that 1+a*b=(1+a)(1+b).

not that G=R\{-1} means that G is all real numbers excluding -1.

Originally Posted by TexasGirl

Show that a*b=a+b+ab defines a group structure on G=R\{-1} using that 1+a*b=(1+a)(1+b).

not that G=R\{-1} means that G is all real numbers excluding -1.

Binary Operation is Closed: The only case where it would not be closed is when,

Thus,

Thus,

Thus,

Impossible, thus the binary operation is closed.

Associative: We have,

And,

Thus,

Identity: We have, that,
and,
.

Inverse: Notice that,

Finally, we need to show if, then, the element
Assume, it were then,

Thus,

Thus,

Thus,

Impossible, thus,