 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/b/b2/b/solution

The cyclic group of order six has two different composition series:

1 ⊲ C_{2} ⊲ C_{6}, 1 ⊲ C_{3} ⊲ C_{6} .

Their factor lists are

C_{2}, C_{3}

and

C_{3}, C_{2}

respectively; normality follows from commutativity.

The Jordan–Hölder theorem states that any two composition series of a group have the same length and the same multiset of composition factors up to isomorphism. Equivalently, the factors of one series can be reordered to match those of the other by group isomorphisms. The actual intermediate subgroups need not be the same.

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