The Jordan-Hölder Theorem
نویسنده
چکیده
The goal of this article is to formalize the Jordan-Hölder theorem in the context of group with operators as in the book [5]. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library. Next, the article formalizes the Zassenhaus butterfly lemma and the Schreier refinement theorem, and defines the composition series.
منابع مشابه
The Kolmogorov continuity theorem, Hölder continuity, and the Kolmogorov-Chentsov theorem
Because X is a modification of Y , the right-hand side is a union of finitely many P -null sets, hence is itself a P -null set. A and B each belong to F , so P (A4B) = 0. Because P (A4B) = 0, P (A) = P (B), i.e. P (Xt1 ∈ A1, . . . , Xtn ∈ An) = P (Yt1 ∈ A1, . . . , Ytn ∈ An). 1We have not assumed that (Ω,F , P ) is a complete measure space, so we must verify that a set is measurable before spea...
متن کاملAnalogues of the Jordan–hölder Theorem for Transitive G-sets
Let G be a transitive group of permutations of a finite set Ω, and suppose that some element of G has at most two orbits on Ω. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and the same sequence of relative indices between consecutive groups (up to permutation). We also deduce the same conclusion when G has a transitive quasi-Hamilton...
متن کاملThe 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 کاملPerturbations of Jordan higher derivations in Banach ternary algebras : An alternative fixed point approach
Using fixed pointmethods, we investigate approximately higher ternary Jordan derivations in Banach ternaty algebras via the Cauchy functional equation$$f(lambda_{1}x+lambda_{2}y+lambda_3z)=lambda_1f(x)+lambda_2f(y)+lambda_3f(z)~.$$
متن کاملA unique common fixed point theorem for six maps in g-metric spaces
In this paper we obtain a unique common xed point theorem for sixweakly compatible mappings in G-metric spaces.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008