Classification of possible multisets of composition factors for groups of a given order
Goal of the classification
We are given a natural number and are asked to determine all the possibilities for the multiset of composition factors (i.e., the isomorphism classes of composition factors with multiplicities) of groups of order . In other words, our goal is to classify the equivalence classes of groups of order under composition factor-equivalence.
Facts used in the classification
- Find the prime factorization of .
- Find all the simple non-abelian groups of orders dividing . You don't actually need to consider all divisors.
- Find all possible multisets of simple non-abelian groups drawn from the collection obtained in (2) for which the product of orders in the multiset divides . For each such multiset, complete it by choosing groups of prime order for the remaining composition factors. Note that one case of special interest is the empty multiset, which when completed, gives a multiset where all pieces are groups of prime order. This particular multiset corresponds to the finite solvable group.
- These completed multisets are precisely the possible multisets.
Note on the composition factor multiset for solvable groups
There is one equivalence class that comprises precisely the finite solvable groups of order . The composition factor multiset for this equivalence class is described as follows: the composition factors are the groups of prime order for primes dividing , and the multiplicity of a particular composition factor is the exponent of the corresponding prime in the prime factorization of . Explicitly, if has the prime factorization:
then there are copies of the composition factor that is the cyclic group of order . The total composition length is .
Note further that this is the unique equivalence class (i.e., there is only one multiset of composition factors) for a given if and only if is a solvability-forcing number.