Question on Functions

[Sonal7]

I have a question on groups and Cayley table which involves functions. I am not sure how to find the function that maps onto the f^-2?
The rest is explanatory and intuitive.
Please see question

 

[pka]

I have a question on groups and Cayley table which involves functions. I am not sure how to find the function that maps onto the f^-2? The rest is explanatory and intuitive.

Frankly, I am not at all sure what it is that you are asking.
It appears that you have completed the Cayley Table correctly.
From that we know that \(\displaystyle h\) is the identity and \(\displaystyle f~\&~g\) are the inverses of each other.
The fact that the table is symmetric about its diagonal tells us what?

 

[Sonal7]

Hi Thank you. I am unsure about part V. I am able to simplify g¹⁰, but unsure how to simple the rest of the function. Ah I see, Its not a suppose to mean 1/f², rather they mean inverse! Such different meanings! I think I might get there now.

 

[pka]

Hi Thank you. I am unsure about part V. I am able to simplify g¹⁰, but unsure how to simple the rest of the function. Ah I see, Its not a suppose to mean 1/f², rather they mean inverse! Such different meanings! I think I might get there now.

I also find the notation very odd indeed.
Had the instructions not said to show it is a group with respect to composition. I would not known that from \(\displaystyle f^2(x)\).
What is \(\displaystyle (\forall n\ge 2)[f^n(x)]~?\)

 

[Steven G]

Hi Thank you. I am unsure about part V. I am able to simplify g¹⁰, but unsure how to simple the rest of the function. Ah I see, Its not a suppose to mean 1/f², rather they mean inverse! Such different meanings! I think I might get there now.

So what did you get for g¹⁰f^-2

 

[Sonal7]

I cheated a bit. But I think this is the way to find the right answer. Might it be that g cube is h, and inverse of f is g. g squared is f. Therefore this simplifies g times f. this is h. The answer I think is h.

 

[Steven G]

I cheated a bit. But I think this is the way to find the right answer. Might it be that g cube is h, and inverse of f is g. g squared is f. Therefore this simplifies g times f. this is h. The answer I think is h.

g¹⁰f^-2= (f²)¹⁰f^-2=f^2*10-2=f¹⁸=h⁶ = h