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

作者：Czedli Gabor^*; Schmidt E Tamas

来源：Algebra Universalis, 2011, 66(1-2): 69-79.

DOI：10.1007/s00012-011-0144-1

• Jordan-Holder theorem
• semimodular lattice
• slim lattice
• planar lattice
• group
• composition series
• subnormal subgroup
• chief series

For subnormal subgroups A del B and C del 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 del Y such that AY = B, A boolean AND Y = X, CY = D, and C boolean AND Y = X. Clearly, B/A congruent to D/C in this case. As G. Gratzer and J. B. Nation have recently pointed out, the standard proof of the classical Jordan-Holder theorem yields somewhat more than is 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.

• 出版日期2011-10