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Yes. You could ask the newer members to not enter those and threaten to kick them otherwise. But they would be able to enter and even stay in the giveaway after having been kicked.

Such a feature like that dosn't exist.
Groups have their own rules, so you may see groups that's ask to not join any ga until your first ga end. And similar. But that are rules, manually checked by mod of the group.
I think you are asking for something more "automated" and no, there is no such feature.

what group u talking about? i wanna join :)

If you want to do that, you have to set the group to private on Steam, do not accept any new members for some time, do the giveaway and set the group to public on Steam afterwards.

The only way would be to write automation scripts that gather info about giveaways and their participants, then provide you information in case you want to kick people out if they haven't followed the rules..

I guess one way would be to make the GA invite-only and to send an invite to the members of the group.
This way you won't get people joining the group just to enter the GA since they won't know what it is or that it's for that group?