Answer to Question #249909 in Abstract Algebra for Israr

Question

Check whether R is a group under binary operation

a*b=a+b-ab

Answer 1

(R,*) is a group, if:

1) $\forall a,b\in R: a*b\in R$

2) $\forall a,b\in R: (a*b)*c=a*(b*c)$

3) $\exist e\isin R$ $\forall a\in R : a*e=e*a=a$

4) $\forall a\isin R$ $\exists a^{-1}\isin R: a*a^{-1}=a^{-1}*a=e$

1) $\forall a,b\in R: (a+b\isin R, ab\in R) \to a*b \in R$

2) $\forall a,b\in R: (a*b)*c = (a+b-ab)*c = a+b-ab+c - (a+b-ab)c =$

$= a+b-ab + c -ac-bc+abc = a-ab-ac+abc+(b+c-bc) =(a+(b+c-bc)) - a(b+c-bc) = a*(b*c)$

3) $a*e = a+e-ae=e+a-ea=e*a$

a * e = a

a + e - ae = a

e(1 - a) = 0

e = 0

$\exist (e = 0)\isin R$ $\forall a\in R : a*0=0*a=a$

4) $a*a^{-1} = e$

$a+a^{-1}-aa^{-1} = 0$

$a+a^{-1}(1-a) = 0$

$a^{-1} = -{\frac a {1-a}}$

For a=1 there is no $a^{-1}$, so R is not a group under binary operation *.