Hi Shirint,
In adCenter, it is impossible to have the ad groups (or campaings) that have the same name. The adCenter web UI will indicate that you are trying to create an item that already exists.
You can, however, have identical ads across multiple ad groups and campaigns. The system will treat each of the ads (in different campaigns/ad groups) as the same as they do have the same ad ID. Since these ad ID's are the same, you will only need to update 1 ad and this will update all ads with the same ad ID.
If you are finding that you do have identical ad group names I would encourage you to contact support as this is a bug that will need to be corrected immediatly.
Thank you,
Brian H - MSFT