The whole point of relational databases is to model your data using tables, then join those tables using keys as others have shown. There are so many reasons why you want to do this, here are just a few of them:

• Sooner or later you will want to define properties for those groups, so you need a table for them anyway
• Using separate tables you are not limited to asking "what groups is this user member of", you can just as easily ask "what users are member of this group"
• As the data set grows, storing the group names over and over again is a complete waste
• What if you want to rename or delete a group at some point in the future
• What if at some point in the future you want to make the groups hierarchial and then for some reason use the same name in several places

The list just goes on...

Learn proper SQL data modeling right from the start, it will save you SO much extra work.