Use of g key in vim normal mode

In vim's normal mode, the g prefix is used for a number of commands. Some commands go somewhere in the document, but other commands deal with file encodings and swapping upper/lower case letters.

• ga - show character encoding
• 10gg - go to line 10
• gg - go to line 1
• gH - start Select line mode
• gr{char} - virtual replace N chars with {char}

What is the missing connection between all these commands?

There's no greater connection to g-commands: it's a mixed bunch. It is an easy prefix and the unbound keys were getting extinct so the less-used maps found a good place behind g.

Simply you're talking about two different things. In some cases g is the short way of "global" (for range command for example), for line moving the g stands for goto.

In VIM commands are often shortened for quick of use.

:help global may help

Btw: for line navigation I've always used the :<lineno> syntax.