my $j = $i * $g;
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This is what is wrong (as explained). The nested c-style loops can produce desired output if you do it like this:

use strict;
use warnings;

my $group = 20;

for ( my $id = 0; $id < 120; $id += $group ) {
    for ( my $grp = 1; $grp <= $group; $grp += 1 ) {

        printf "ID: %d\tGRP: %d\n",
          $id + $grp,
          $grp;
    }
}
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