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I have my GTA V copy for PS3, but I have an Xbox 360 as well. If someone lends me the game for Xbox, can I install it and use my PS3 Social Club account for online mode in the Xbox 360?
Yes, the Social Club account is device independent, as long as each device uses the same login details for the account you play the game with.
It's obviously just the Social Club that is shared though, not achievements and game progress.
Required, but never shown