In the set of real numbers R there is an operation defined as: x * y = 3 (x3+y3)2. prove that (R,x) is an Abelian Group.
Answers
Assume (G,∗,e) is any group with a binary operation
∗: G×G→G and neutral element e, and H is a set that admits a bijection
h: H -> G.
Then we may define a binary operation
□: H×H→H by a□b:=h−1(h(a)∗h(b))
which makes H a group with neutral element f:=h−1(e): For any an∈H we have by definition
a□f = h − 1 (h (a) ∗ h (f)) = h − 1 (h (a) ∗ e ) = h − 1 (h (a) ) = a ,
and along the same lines also f□a=a , Thus f is the neutral element of H . Now ,
b=h−1(h(a)inv)
is indeed the inverse element of a∈H , where I have denoted taking an inverse element in G with respect to the group operation by the subscript inv in order to avoid confusion with the use of the superscript −1 for the inverse map h−1 :
a□b = h − 1 (h (a) ∗ h (b)) = h − 1 (h (a) ∗ h (a) inv) = h − 1 (e) = f ,
and along the same lines also b□a=f is shown. It is left to the reader to verify the binary operation □ on His associative. Thus (H,□,f) is a group.
The group structure on G is transferred to H via the bijection h: H -> G, and by construction H is isomorphic to G . Note that the set H is not assumed to be endowed with any structure before lifting the group structure of G to H.