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Depends on how recoil sensitive your girlfriend is.
I wouldn't hesitate to hunt elk or moose with my 6.5 x 55 and the recoil is far less than a 30-06.
If you get the Ruger American, I'd recommend going with the 270. That's a relatively light rifle and recoil will be pretty bad with the 30/06.With the 270 you can go with a lighter 130 grain bullet, and it's plenty for elk, especially if you use good bullets like Nosler Accubonds or Barnes.