Corrections : G is a group of functions from R to R-star, IE from R to set of non zero real numbers. So to be precise we can define identity function as e(x):= x when x is a non zero real And e(x):= 1 when x is zero But then e(x) will not be invertible hence we should consider functions from R-star to R-star.
1)I think the said group G of functions is under 'Multiplication' operation for which identity element is e(x)=1. (2)On the other hand group G of functions under 'Composition' operation will have identity function e(x)=x as its identity element... (3) e(x)=1 is a kind of 'constant function' of the form e(x)=k. I think this function maps from R to R*. 4) While identity function e(x)=x cant map from R to R* as you mentioned above
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Corrections :
G is a group of functions from R to R-star, IE from R to set of non zero real numbers.
So to be precise we can define identity function as
e(x):= x when x is a non zero real
And e(x):= 1 when x is zero
But then e(x) will not be invertible hence we should consider functions from R-star to R-star.
Very nice lecture
1)I think the said group G of functions is under 'Multiplication' operation for which identity element is e(x)=1.
(2)On the other hand group G of functions under 'Composition' operation will have identity function e(x)=x as its identity element...
(3) e(x)=1 is a kind of 'constant function' of the form e(x)=k. I think this function maps from R to R*.
4) While identity function e(x)=x cant map from R to R* as you mentioned above