Any error messages ?
As a first suggestion , download the latest Chrome browser and start that.
I can see no reason why Google Chrome.app would not run on Mavericks.
Just right click on the Chrome file and select open. That should work and it will install. It's a safe program and will run OK on your new iMac. The reason it wouldn't install is because of the built in security protection in Mavericks. Programs downloaded from other than the Mac App store will refuse to install for the same reason.
Shop for your Apple, Mac, iPhone and other computer products on Amazon.
We are a participant in the Amazon Services LLC Associates Program, an affiliate program designed to provide a means for us to earn fees by linking to Amazon and affiliated sites.