sp_send_dbmail can only send to an address that conforms to an actual email address format, ie email@example.com; you cannot send to just "MyGroup"
a group account is handled special by an exchange server; it is optional to expose the group to external resources outside of the domain, so you have to get the mail admin to map firstname.lastname@example.org; to your group; when the exchange mail server receives the email, it will process it and duplicate the email out to each of the real members of the group.
get with your Exchange admin to create that or to make sure you know the mapped email for it.
it's optional specifically because it would be too easy to get spam from outside world with a email@example.com going out to everyone, for example.
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