The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices

For subnormal subgroups A / B and C / D of a given group G, the factor B/A will be called subnormally down-and-up projective to D/C, if there are subnormal subgroups X /Y such that AY = B, A∩Y = X , CY = D and C∩Y = X . Clearly, B/A ∼= D/C in this case. As G. Grätzer and J.B. Nation [6] have just pointed out, the standard proof of the classical Jordan-Hölder theorem yields somewhat more than widely known; namely, the factors of any two given composition series are the same up to subnormal down-and-up projectivity and a permutation. We prove the uniqueness of this permutation. The main result is the analogous statement for semimodular lattices. Most of the paper belongs to pure lattice theory; the group theoretical part is only a simple reference to a classical theorem of H. Wielandt [14].