Thank you for trusting us with your question.
Is the groan your'e hearing more of a whing noise, and did it just start, or has it been doing this for a while, gradually getting louder?
Typically what causes this, is wear in the toilet valve, causing
a small part of the valve to vibrate as water passes thru it.
Usually the only way to stop it is to replace the valve.
This is not a difficult job.
Make sure to completely drain the toilet tank, or you will have a big mess.
Turn the water to the toilet off and disconnect the line from the bottom of the toilet, you should find a large nut on the pipe.
Make sure the tank is completely before removing this nut.
Remove this nut, and the whole valve assembly should come out.
Then just take the whole thing to your local hardware store, and the should be able to fix you up with a replacement. (theyr'e pretty standard)
Reverse the process and it should be good.
I hope this helps you and if you need further assistance from me please feel free to ask.
Thank you again for trusting us with your question.
Ok, please let me know if you need anything further