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On the maps implicit in the Jordan-Hölder theorem. (English) Zbl 0905.20018
Hofmann, Karl H. (ed.) et al., Semigroup theory and its applications. Proceedings of the 1994 conference commemorating the work of Alfred H. Clifford, New Orleans, LA, USA, March 1994. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 231, 95-106 (1996).
As follows from the Jordan-Hölder theorem, for any two composition series \(A_0,A_1,\dots,A_n\) and \(B_0,B_1,\dots,B_n\) of a group \(G\) there is a permutation \(\pi\) of the set \(\{1,2,3,\dots,n\}\) such that, for each \(i=1,2,3,\dots,n\), \(A_i/A_{i-1}\approx B_{\pi(i)}/B_{\pi(i)-1}\). Any proof of the Jordan-Hölder theorem actually constructs the permutation \(\pi\) and the corresponding isomorphisms become explicit. But for different proofs there are different choices of \(\pi\) or the isomorphisms. The main goal of the article under review is to examine the extent to which these maps may vary.
For the entire collection see [Zbl 0890.00035].
20E15 Chains and lattices of subgroups, subnormal subgroups
20F14 Derived series, central series, and generalizations for groups