ITunes won't let me transfer music from?

I erased & synced on a friends comp so all her music is on my iphone. My assumption was that the original comp my iphone was synced to would just let me re-enter its music onto my iphone but it doesn't. So I authorized my comp so it would 'transfer purchases from my phone' (an option in the file menu) and it said it was successful but the music doesn't show up in my itunes music. Any ideas on what went wrong?

You may want to try removing DRM from iTunes music files to use them on any device?
Do you? Try Requiem (Free DRM Removal Tool)

Requiem 3.3.5 for DRM Removal which is latest version and compatible with Mac & Windows and able to remove the protection on media content (Music and Movies) taken from iTunes, and the protection on iBooks as well.

You can get it for free with tips & steps at that site
http://www.ismoothblog.com/2012/02/bypass-apple-ibooks-drm-protection.html

You should authorize your computer with your friends' Apple IDs, otherwise you could not play their songs on your computer.

There's another way that you can use a DRM Converter like Noteburner Audio Converter (http://www.noteburner.com/noteburner-audio-converter.html) to remove the DRM from your friends songs on their computers, then transfer them to your iphone, then you can put the DRM-free songs to your computer without any restrictions.

Hope this helps!