Given the set G=(0,1) and the law of composition x*y=xy/(2xy-x-y+1), Verify if e=1/2 is the neutral element of the group G.

The property of a neutral element of a group
is:

e*g = g*e = g, where g is an element of the set
G.

We'll apply the composition law and we'll replace e by
1/2:

x*(1/2) = x

We'll replace
x*(1/2) by x/2/(2x/2 - x - 1/2 + 1)

We'll compute the
expression to check if it yields x.

x/2/(2x/2 - x - 1/2 +
1) = x/2(x - x - 1/2 + 1)

x/2(1 - 1/2) = x/(2/2) = x/1 =
x

We notice that the expression x*(1/2) = x,
therefore e = 1/2 represents the neutral element of the given group, whose law of
composition is x*y=xy/(2xy-x-y+1).