How to get first record in each group using Linq

var result = input.GroupBy(x => x.F1, (key,g) => g.OrderBy(e => e.F2).First());

Use it to achieve what you want. Then decide which properties you want to return.

yourList.OrderBy(l => l.Id).GroupBy(l => new { GroupName = l.F1}).Select(r => r.Key.GroupName)

var res = from element in list
          group element by element.F1
              into groups
              select groups.OrderBy(p => p.F2).First();

The awnser of @Alireza is totally correct, but you must notice that when using this code

var res = from element in list
          group element by element.F1
              into groups
              select groups.OrderBy(p => p.F2).First();

which is simillar to this code because you ordering the list and then do the grouping so you are getting the first row of groups

var res = (from element in list)
          .OrderBy(x => x.F2)
          .GroupBy(x => x.F1)
          .Select()

Now if you want to do something more complex like take the same grouping result but take the first element of F2 and the last element of F3 or something more custom you can do it by studing the code bellow

 var res = (from element in list)
          .GroupBy(x => x.F1)
          .Select(y => new
           {
             F1 = y.FirstOrDefault().F1;
             F2 = y.First().F2;
             F3 = y.Last().F3;
           });

So you will get something like

   F1            F2             F3 
 -----------------------------------
   Nima          1990           12
   John          2001           2
   Sara          2010           4