Hello,

learning for my algebra exam I came across a task I cannot wrap my head around for some time already.

The task is to decide weather formula (not sure with English terminology here, sorry)(f * g)(r) = f(g(r) - 1), forf,g: R -> Randr \in R, defines operation on setSof all injective real function, such that(S,*)is semigroup, group respectively.

So far I what I've done is that I had shown that*is a function fromS^2 -> Slike so:

Screen Shot 2018-01-15 at 02.52.34.jpg

Next, I've shown, using the functionhfrom above that * is associative:

Screen Shot 2018-01-15 at 02.58.31.png

which should make(S, *)a semigroup.

Then I found a neutral elemente(r)=r+1, r \in R :Screen Shot 2018-01-15 at 03.01.44.png

existence of which should imply(S,*)is a monoid.

Lastly, to decide whether it's group I need to decide whether every element inShas an inverse (meaning that for each f \in S there is som g \in S that f*g = e and g*f = e) or not. This is where/when my brain stopped working (partially because it's 3 AM here but I cannot sleep without finishing this one). I cannot find way to show that there is such g for any f, nor the opposite.

Any hints, please?

Thank you.

P.S. this is my first post here so I hope I formatted the question OK. Have a nice day all of you.

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