If the group is type Open then that'll be the case. If the group type is Approval it requires the group owner, group admin, or group moderators to approve the join request. If it's type Invite then they can only join if they've been invited by another user.
Further question: Is it possible to choose a group during the reistration. After the registration the new user will be a member of the group automatically (only a information will be sent to the admin).
Yes, install GJ Auto and create an Auto Join field in CB > Field Management.