When I send a message to my email alias, or a group I belong to, it doesn't show up in my inbox.
Example: John's primary email address is email@example.com. He also has the email alias firstname.lastname@example.org. If John includes email@example.com as a recipient of a message, the message does not appear in his inbox. However, if John sends a message to his primary account (firstname.lastname@example.org), then the message appears in his inbox.
To prevent clutter, Gmail doesn't route 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. If you need to see messages sent to your alias in the Inbox, you can configure the alias as an alternate "Send mail as" address for your account.