The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices
نویسندگان
چکیده
منابع مشابه
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 wi...
متن کاملThe Jordan-Hölder Theorem
This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theo...
متن کاملAn extension theorem for planar semimodular lattices
We prove that every finite distributive lattice D can be represented as the congruence lattice of a rectangular lattice K in which all congruences are principal. We verify this result in a stronger form as an extension theorem.
متن کاملFrankl’s Conjecture for Large Semimodular and Planar Semimodular Lattices
A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote...
متن کاملSemimodular Lattices and Semibuildings
In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i . Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Hölder permutation” between any two flags. This permutation has the properties of an Sn-distance function on the chamber system of flags. Using these noti...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Algebra universalis
سال: 2011
ISSN: 0002-5240,1420-8911
DOI: 10.1007/s00012-011-0144-1