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Groups question - showing that a set forms a group?
Show that the set of real numbers excluding -1 forms a group under the operation * defined by a*b = a +b + ab
Thanks so much!
1 Answer
- GusBsAsLv 61 decade agoFavorite Answer
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).
