11.10 - Synaptic package manager is missing

There is no synaptic package manager in 11.10,how should i completely remove(Purge) an installed application?

Does clicking on "install" on any software in Ubuntu Software centre means that the command

sudo apt-get install XXXXXXX

will be used in the background?

You can always install Synaptic from Ubuntu Software Center. From there, you can purge.

sudo apt-get purge will also do the trick from the command line. See man apt-get.