Here’s an explanation of how the default example for LINQ nested grouping actually works. The usual example for nested grouping looks like this:

from student in Students
group student by student.Faculty into Faculty
from dbtgroup in
(
    from student in Faculty
    group student by student.DebtCategory
)
group dbtgroup by Faculty.Key;

The objective of this statement is to first group-by students into faculties and then in each faculty create subgroupings of students by their DebtCategory.

So how does this actually work and whats the equivalent method/lamba syntax? The first step is to groups each student into their faculty. Assume we have the following data

public class Student
{
   public string Name { get; set; }
   public string Faculty { get; set; }
   public int DebtCategory { get; set; }
}

IList<Student> Students = new List<Student>();
Students.Add(new Student { Name = "John" , Faculty = "IT"     , DebtCategory = 2 });
Students.Add(new Student { Name = "Jane" , Faculty = "IT"     , DebtCategory = 2 });
Students.Add(new Student { Name = "Jesse", Faculty = "Finance", DebtCategory = 2 });
Students.Add(new Student { Name = "Linda", Faculty = "Finance", DebtCategory = 1 });

The following query groups each student into a faculty

var query1 = from student in Students
group student by student.Faculty into Faculty
select Faculty;

//The Method syntax for the above query is:
var query1Method = Students
.GroupBy(student => student.Faculty)
.Select ( Faculty => Faculty);

//This gives us the following IGrouping<string, Student> as result
//
// [0]
//    Key   :  IT
//    Values: 
//          [0] John (IT) (2)
//          [1] Jane (IT) (2)
//
// [1]
//    Key   : Finance
//    Values:
//          [0] Jesse (Finance) (2)
//          [1] Linda (Finance) (1)

The next step is to add another level of grouping:

var query2 = from student in Students
group student by student.Faculty into Faculty
from dbtgroup in
(
    from student in Faculty
    group student by student.DebtCategory
)
select dbtgroup;
//This gives us the following IGrouping<int, Student> as result
//[0]
//  Key   : 2
//  Values:
//        [0] John (IT) (2)
//        [1] Jane (IT) (2)
//
//[1]
//  Key   : 2
//  Values:
//        [0] Jesse (Finance) (2)
//
//[2]
//  Key   : 1
//  Values:
//        [0] Linda (Finance) (1)

// The following is the literal translation of the above Comprehension syntax into method syntax. We're ignoring this as explained below
//	var query2Method = Students
//		.GroupBy(student => student.Faculty)
//		.SelectMany(  Faculty =>Faculty.GroupBy(student => student.DebtCategory)
//					, (Faculty, dbtgroup) => dbtgroup);
	
//The final complete query ends with"group dbtgroup by Faculty.Key;" 
// this statement causes the compiler to see that you're refering to the Faculty object from the select many, so instead of 
// "(Faculty, dbtgroup) => dbtgroup" it emits a slightly different projection "(Faculty, dbtgroup) => new {Faculty, dbtgroup}
//structure
var query2Method = Students
.GroupBy(student => student.Faculty)
.SelectMany( Faculty =>Faculty.GroupBy(student => student.DebtCategory)
	 , (Faculty, dbtgroup) => new {Faculty, dbtgroup});

Query2 is close to our desired output, however the grouping is the wrong way around. So the final step is:

var query3 = from student in Students
group student by student.Faculty into Faculty
from dbtgroup in
    (
    from student in Faculty
    group student by student.DebtCategory
    )
group dbtgroup by Faculty.Key;

//The method/lambda syntax is:
var query3Method = Students
.GroupBy(student => student.Faculty)
.SelectMany (
	Faculties => Faculties.GroupBy (student => student.DebtCategory)
	, (Faculty, dbtgroup) => 
		new  
		{
			Faculty = Faculty, 
			dbtgroup = dbtgroup
		} )
.GroupBy( item => item.Faculty.Key, item => item.dbtgroup );

//This gives us the following groups as result
//[0]
//  Key   : IT
//  Values:
//        [0] Key   : 2
//            Values:
//                  [0] John (IT) (2)
//                  [1] Jane (IT) (2)
//[1]
//  Key   : Finance
//  Values:
//        [0] Key   : 2
//            Values:
//                  [0] Jesse (Finance) (2)
//        [1] Key   : 1
//            Values:
//                    [0] Linda (Finance) (1)