What happens if you right click on that F icon? If you don't have a 2 button mouse, go control+click. Do you see Get Info? If so click on that and let us know what you see. Just a guess.
I am a newbe with iMac and therefore I take all the responses to my query serious and I am grateful for the various suggestions. I will try to give answers to each response on the subject.
Control+Click does not do anything. However a double click will open a window.This is an indication that the Icon in question is legitimate and was deliberately placed on the desktop. My problem is that I just cannot remove it. Facebook might be good for a lot of people, but does nothing useful for me other than annoy me with a multitude of silly conversations with very little intelligent content.
But lets examine the functionality of this app. It permits to create messages to two different destinations. When pressing the "profile" button a list of my facebook 'friends' shows up which would allow me to send a message to the Wall of a selected friend. If pressing the "post" button instead will send the messig onto my own Wall.
So, there is nothing wrong with the intended funcionality. Facebook has a very agressive way to extend its user base. I don't like this agressivenes when it affect the working of my computer. I should be able to remove the app like any other app. But unfortunately it does not and I am trying to find out why not.
John