As the title asked. I just bought a Authenticator now .Can I still get a pet ??

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Q u o t e:
Yep if you've activated it. You should get an in game mail.

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Exactly correct. From the following page: http://us.blizzard.com/support/article.xml?locale=en_US&articleId=24660

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Q u o t e:
How do I get the special Core Hound Pup pet for Authenticator users?

If you have a Battle.net Authenticator or Battle.net Mobile Authenticator attached to your Battle.net account, you'll find the Core Hound Pup item waiting for you in your in-game mailbox the next time you log in. Simply use the item to add it to your pet collection.

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Q u o t e:
Does the pet get sucked into the nether alone and scared if you take the authenticator off your account?

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From the same link I gave ya up above:

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Q u o t e:
What if I remove my Authenticator?

If you remove the Authenticator from your Battle.net account, the Core Hound Pup will disappear from the pets collection of all of your characters after you log out of the game. You'll need to reattach an Authenticator to receive the pet again, at which point you'll find him in your mailbox once more. If you switch Authenticators while in-game, your pet will remain intact.

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