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my %hGroups = (
          'Group1' => {
...
                        'ATG1' => 1
                      }
        );

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my %hGroups = map { my $sGenes = $hash{$_};
                    my $hGroupMembers
                      = { map { $_ => 1 } split(',', $sGenes) };
                    $_ => $hGroupMembers;
                  } keys %hash;

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my %hNewGroups;

...

  $hNewGroups{$sGroup} = $hGroupMembers;
}

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%hNewGroups = (
          'Group1' => {
...
                        'LSM2' => 1
                      }
        );

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while (my ($sNewGroup,$hMembers) = each(%hNewGroups)) {
  print "$sNewGroup: " . join(',', keys %$hMembers) . "\n";
}

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Group1: ATG9,ATG2,ATRG7,ATG4,ATG1
Group3: FYCO1,LSM2