Groups question - showing that a set forms a group?

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G1) (a*b)*c = a*(b*c) for any a,b,c in A=R\{-1}

(a*b)*c = (a+b+ab)c = a+b+ab + c + (a+b+ab)c =
= a+b+c + ab+ac+ab+ abc

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

Then a*(b*c) = (a*b)*c

G2) There exists u in A so that a*u = u*a = a for every a in A.

a*u = a + u + au = a
=> u + au = 0
=> u (1+a) = 0 for every a not equal to -1.

Then u=0

a*0 = a+0+a*0 = a, ok.

G3) For any a in A, there exists b in A so that a*b = b*a = u

a*b = a+b+ab = u = 0
=>
b(1+a) = -a
=>
b = -a/(1+a)

Such b exists because a in not equal to -1.

Therefore, (A,*) is a group (moreover, it is a conmutative group)....Show more