As explained in the documentation about group members there are multiple ways someone can become part of a group:
1. An administrator can add someone to a group on the admin site of Groups
Upon adding them, the Groups administrator can select whether they should be notified by email or not.
This method does not send the new member an email notification.
3. A person can request membership on the group’s public page.
Any visitor can request to join a public/open group on the group’s group page. A Groups administrator or the group leader can approve this request, making the person a member.
When a new membership request is submitted, an email will notify the group leaders of the pending request. Group leaders can review all outstanding membership requests at the bottom of their group page here:
Outstanding membership requests will also be visible on the administrative side of Groups, on the notifications tab: