Maximal class group: Difference between revisions

Revision as of 19:22, 25 July 2009

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

A maximal class group is a group of prime power order that has nilpotency class $c$ , and order $p^{c + 1}$ , where $p$ is a prime number and $c > 1$ . Equivalently, it has the following properties:

The abelianization of the group, i.e., the quotient of the group by its commutator subgroup has order $p^{2}$ .
The upper central series and lower central series coincide, and all the successive quotients (except the top-most quotient) are of order $p$ .

Relation with other properties

Weaker properties

UL-equivalent group

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 ==Definition==
-A '''maximal class group''' is a [[group of prime power order]] with the following special structure:
+A '''maximal class group''' is a [[group of prime power order]] that has [[nilpotency class]] <math>c</math>, and order <math>p^{c+1}</math>, where <math>p</math> is a prime number and <math>c > 1</math>. Equivalently, it has the following properties:
-* The group has [[nilpotence class]] <math>c</math>, and order <math>p^{c+1}</math>, where <math>p</math> is a prime number
+# The [[abelianization]] of the group, i.e., the quotient of the group by its [[commutator subgroup]] has order <math>p^2</math>.
-* The quotient of the group by its [[commutator subgroup]] has order <math>p^2</math>
+# The [[upper central series]] and [[lower central series]] coincide, and all the successive quotients (except the top-most quotient) are of order <math>p</math>.
-* The [[upper central series]] and [[lower central series]] coincide, and all the successive quotients (except the top-most quotient) are of order <math>p</math>
 ==Relation with other properties==