Yes, you can. Sometimes it’s useful for people to use a group email list without knowing the identities of other members of the group. If you enable ‘Incognito mode’ in your group settings people can participate in the group without their email address being visible.
When someone joins a group they can optionally enter their name, to preserve anonymity it is better to enter a name; either abbreviated (i.e. Bob) or fake, otherwise Gaggle Mail will extract their name from emails sent to the group and show that.