Para agregar opción desplegable usando jQuery
$('#ddlGroup').append(new Option('Select group', '0'));
Light Leopard
$('#ddlGroup').append(new Option('Select group', '0'));BY LOVE
$.each(obj , function (key, value) {
$('#GroupName').append($('<option>',
{
value: value.id,
text: value.Group_Name
}));