When I send a message to my email alias or to a Group I belong to, the message isn't sent to my inbox.
Example: John's primary email address is firstname.lastname@example.org. He also has the email alias email@example.com. If John includes firstname.lastname@example.org as a recipient of a message, the message isn't sent to his inbox. However, if John sends a message to his primary account, email@example.com, the message is sent to his inbox.
To prevent clutter, Gmail doesn't deliver messages that you send to your own alias (or to a Group you belong to) to your inbox. You can find the message in Sent Mail or All Mail. To get these messages in your inbox, set up the alias or Group as an alternate "Send mail as" address for your account.