Difference between revisions of "Jordan-Dedekind lattice"
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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]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan ( | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1964) {{ZBL|0116.25403}}</TD></TR> | ||
| + | </table> | ||
[[Category:Order, lattices, ordered algebraic structures]] | [[Category:Order, lattices, ordered algebraic structures]] | ||
Latest revision as of 19:05, 1 May 2024
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 (1964) Zbl 0116.25403 |
How to Cite This Entry:
Jordan-Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan-Dedekind_lattice&oldid=33658
Jordan-Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan-Dedekind_lattice&oldid=33658
This article was adapted from an original article by L.M. Batten (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article