Difference between revisions of "Jordan-Dedekind lattice"
From Encyclopedia of Mathematics
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A [[Lattice|lattice]] satisfying the following condition, known as the Jordan–Dedekind chain condition: All maximal chains have the same length. | A [[Lattice|lattice]] satisfying the following condition, known as the Jordan–Dedekind chain condition: All maximal chains have the same length. | ||
| − | The condition arose in connection with the Jordan–Hölder theorems for groups (cf. [[Jordan–Hölder theorem|Jordan–Hölder theorem]]), and is equivalent to the condition of supersolvability in the lattice of all subgroups of a finite group. | + | The condition arose in connection with the Jordan–Hölder theorems for groups (cf. [[Jordan–Hölder theorem|Jordan–Hölder theorem]]), and is equivalent to the condition of supersolvability in the lattice of all subgroups of a finite group (cf [[Supersolvable group]]). |
A general reference is [[#References|[a1]]]. See also [[Partially ordered set|Partially ordered set]]; [[Chain|Chain]]. | A general reference is [[#References|[a1]]]. See also [[Partially ordered set|Partially ordered set]]; [[Chain|Chain]]. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1968)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1968)</TD></TR></table> | ||
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| + | [[Category:Order, lattices, ordered algebraic structures]] | ||
Latest revision as of 20:22, 14 October 2014
A lattice satisfying the following condition, known as the Jordan–Dedekind chain condition: All maximal chains have the same length.
The condition arose in connection with the Jordan–Hölder theorems for groups (cf. Jordan–Hölder theorem), and is equivalent to the condition of supersolvability in the lattice of all subgroups of a finite group (cf Supersolvable group).
A general reference is [a1]. See also Partially ordered set; Chain.
References
| [a1] | M. Hall, Jr., "The theory of groups" , Macmillan (1968) |
How to Cite This Entry:
Jordan-Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan-Dedekind_lattice&oldid=15519
Jordan-Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan-Dedekind_lattice&oldid=15519
This article was adapted from an original article by L.M. Batten (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article