This version eliminates the unless statement, but is basically the same as your example (you should use strict). I'm running this on a Solaris 8.0 system but I don't get any duplicates.

use strict;

my @sec_gp;
my $name = "xxxxx";
while (my ($gname,$gpasswd,$gid,$members)=getgrent) {
   my @list_members=split (/\s+/,$members);
   my @list=grep (/$name/,@list_members);
   if ($list[0] eq $name) {
      push (@sec_gp,"$gname");
   }
}
print "\@sec_gp: ";
print "$_\n" foreach @sec_gp;
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Output:

@sec_gp: acad libsoftwrite helpdesk codes
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--Jim Update: The above was hastily done. Better is the following (still room for improvement I'm sure):

   push(@sec_gp, "$gname") if $list[0] eq $name;
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for the original if statement.