I have an iMac 3.4 Ghz Intel Core i7 running El Capitan, can I upgrade to a more recent operating system

See if you can upgrade to 10.12, 10.1., see sys requirements...

https://support.apple.com/kb/sp742?locale=en_US

https://itunes.apple.com/us/app/macos-sierra/id1127487414?ls=1&mt=12

https://support.apple.com/kb/sp765?locale=en_US

https://itunes.apple.com/app/macos-high-sierra/id1246284741?mt=12

The iMac 27" mid 2011 can be upgraded to 10.13.6. See: How to upgrade to macOS High Sierra - Apple Support

What model year is your iMac (from Apple menu > About this Mac)?

What year is the iMac?

Yes, cause all iMacs with core i7 cpus supported at least high sierra

Thanks, I was thinking that, but too under the weather to investigate! :)

And the only iMacs that used the 3.4 Ghz i7 were both 27-inch models from Mid 2011 and Late 2012.

Thanks! :)

Great news, take care my friend! :)