Jordan-Hölder: composition series and their factors

Builds composition series for several groups and lists the orders of their simple factors. Different groups of the same order can share the same factors, and the Jordan-Hölder theorem says that every series of one group has the same factors.

from mathematicskit.abstract_algebra import CyclicGroup, DihedralGroup, PermutationGroup, composition_series

Composition factors

groups = {"Z_12": CyclicGroup(12), "D_6": DihedralGroup(6), "S_4": PermutationGroup(4), "Z_24": CyclicGroup(24)}
for name, group in groups.items():
    result = composition_series(group)
    orders = [len(h) for h in result.series]
    print(f"{name}: series orders {orders}, factor orders {result.factor_orders}")

Z_12: series orders [12, 6, 3, 1], factor orders [2, 2, 3]
D_6: series orders [12, 6, 3, 1], factor orders [2, 2, 3]
S_4: series orders [24, 12, 4, 2, 1], factor orders [2, 3, 2, 2]
Z_24: series orders [24, 12, 6, 3, 1], factor orders [2, 2, 2, 3]

Same factors, different groups

Z_12 (abelian) and D_6 (non-abelian) are not isomorphic, yet both have composition factors of orders 2, 2, 3. The Jordan-Hölder theorem makes the factors an invariant of a group, but not a complete one.