Metabelian group: Difference between revisions

Latest revision as of 19:24, 20 June 2013

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

QUICK PHRASES: abelian-by-abelian, abelian normal subgroup with abelian quotient, abelian derived subgroup, abelian commutator subgroup

Symbol-free definition

A group is said to be metabelian or is said to have derived length two or solvable length two if it satisfies the following equivalent conditions:

It is solvable of derived length at most two, i.e., the derived series has length at most two.
Its derived subgroup (i.e., commutator subgroup with itself) is abelian.
There is an abelian normal subgroup that is also an abelian-quotient subgroup, i.e., an abelian normal subgroup with abelian quotient group.
Any two commutators (i.e., elements that can be expressed as commutators of elements of the group) commute with each other.

NOTE: Sometimes, the term metabelian or derived length two or solvable length two is used specifically for a group whose derived length is precisely two, i.e., a non-abelian metabelian group. This is more restrictive than the typical usage of the term.

History

Origin

The concept and term metabelian group was introduced by Furtwangler in 1930.

The term metabelian was earlier used for groups of nilpotency class two, which is a much stronger condition, but is no longer used in that sense.

Formalisms

In terms of the meta operator

This property is obtained by applying the meta operator to the property: abelian group
View other properties obtained by applying the meta operator

The property of being metabelian arises by applying the meta operator to the group property of being Abelian. Equivalently metabelian can be described as Abelian-by-Abelian, where by denotes the group extension operator.

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Generic examples

The trivial group is metabelian; in fact, it has derived length zero.
Any abelian group is metabelian; in fact, it has derived length zero.
Any group of nilpotency class two is metabelian.
Any group arising as the holomorph of a cyclic group (i.e., the semidirect product of a cyclic group and its automorphism group) is metabelian. This is because cyclic implies abelian automorphism group -- the automorphism group of a cyclic group is abelian. More generally, any group arising as the semidirect product of a cyclic group and any subgroup of its automorphism group is abelian.
Any dihedral group is metabelian: it has a cyclic normal subgroup and the quotient group is cyclic of order two. (This is a special case of the previous example). It turns out that dihedral groups are nilpotent only if their order is a power of two; further, the nilpotency class of a dihedral group of order $2^{n}$ is $n-1$ .
Any generalized dihedral group is metabelian: it has an abelian normal subgroup and the quotient group is cyclic of order two.

Particular examples

The symmetric group of degree three, which is also the dihedral group of order six and degree three, is metabelian. It is the smallest example of a metabelian non-abelian group; it is also not nilpotent. The abelian normal subgroup in this case is the alternating group, i.e., the subgroup generated by a $3$ -cycle $\langle (1,2,3)\rangle$ , and the quotient group is cyclic of order two.
The dihedral group of order eight (degree four) is metabelian. We have several choices here of the abelian normal subgroup. We can take the center as the abelian normal subgroup, in which case the quotient is isomorphic to a Klein four-group. Alternatively, we could take as our abelian normal subgroup any of the subgroups of order four.
the alternating group of degree four (order twelve) is metabelian. It has an abelian normal subgroup of order four: the Klein four-subgroup comprising the identity and the three double transpositions $(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)$ . The quotient is a cyclic group of order three.

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
cyclic group		Abelian group\|FULL LIST, MORE INFO
abelian group		\|FULL LIST, MORE INFO
group of nilpotency class two	commutator subgroup is central	\|FULL LIST, MORE INFO
metacyclic group	cyclic normal subgroup, cyclic quotient	\|FULL LIST, MORE INFO
group whose automorphism group is abelian	the automorphism group is an abelian group	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
group with nilpotent derived subgroup	derived subgroup is nilpotent		\|FULL LIST, MORE INFO
abelian-by-nilpotent group	abelian normal subgroup, nilpotent quotient		\|FULL LIST, MORE INFO
solvable group			\|FULL LIST, MORE INFO
group satisfying subnormal join property	join of subnormal subgroups is subnormal	metabelian implies subnormal join property	\|FULL LIST, MORE INFO
metanilpotent group	nilpotent normal subgroup, nilpotent quotient		\|FULL LIST, MORE INFO

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a metabelian group is metabelian. This follows from the general fact that the derived series of the subgroup is contained (entry-wise) in the derived series of the whole group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a metabelian group is metabelian. This follows from the fact that the derived series of the quotient is the quotient of the derived series of the original group.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of metabelian groups is metabelian. This follows from the fact that the derived series of the direct product is the direct product of the respective derived series.

@@ Line 1: / Line 1: @@
+[[importance rank::2| ]]
+{{semibasicdef}}
 {{group property}}
+==Definition==
+{{quick phrase|[[quick phrase::abelian-by-abelian]], [[quick phrase::abelian normal subgroup with abelian quotient]], [[quick phrase::abelian derived subgroup]], [[quick phrase::abelian commutator subgroup]]}}
+===Symbol-free definition===
+A group is said to be '''metabelian''' or is said to have '''derived length two''' or '''solvable length two''' if it satisfies the following equivalent conditions:
+# It is [[defining ingredient::solvable group|solvable]] of [[defining ingredient::derived length]] at most two, i.e., the [[derived series]] has length at most two.
+# Its [[defining ingredient::derived subgroup]] (i.e., commutator subgroup with itself) is [[defining ingredient::abelian group|abelian]].
+# There is an [[defining ingredient::abelian normal subgroup]] that is also an [[defining ingredient::abelian-quotient subgroup]], i.e., an abelian normal subgroup with abelian [[quotient group]].
+# Any two [[defining ingredient::commutator]]s (i.e., elements that can be expressed as commutators of elements of the group) commute with each other.
+{{quotation|'''NOTE''': Sometimes, the term ''metabelian'' or ''derived length two'' or ''solvable length two'' is used specifically for a group whose derived length is ''precisely'' two, i.e., a non-abelian metabelian group. This is more restrictive than the typical usage of the term.}}
 ==History==
@@ Line 7: / Line 24: @@
 The concept and term '''metabelian group''' was introduced by [[Furtwangler]] in 1930.
-The term '''metabelian''' was earlier used for groups of nilpotence class 2, but is no longer used in that sense.
+The term '''metabelian''' was earlier used for [[group of nilpotency class two|groups of nilpotency class two]], which is a ''much stronger condition'', but is no longer used in that sense.
+==Formalisms==
+{{obtainedbyapplyingthe|meta operator|abelian group}}
+The property of being metabelian arises by applying the [[meta operator]] to the [[group property]] of being [[Abelian group|Abelian]]. Equivalently metabelian can be described as Abelian-by-Abelian, where ''by'' denotes the [[group extension operator]].
-==Definition==
+==Examples==
-===Symbol-free definition===
+{{group property see examples}}
-A group is said to be '''metabelian''' if it satisfies the following equivalent conditions:
+===Generic examples===
-* It is [[solvable group|solvable]] of [[solvable length]] 2.
+* The [[trivial group]] is metabelian; in fact, it has derived length zero.
-* Its [[commutator subgroup]] is [[Abelian group|Abelian]].
+* Any [[abelian group]] is metabelian; in fact, it has derived length zero.
-* There is a [[normal Abelian subgroup]] such that the quotient is also Abelian.
+* Any [[group of nilpotency class two]] is metabelian.
+* Any group arising as the [[holomorph of a group|holomorph]] of a [[cyclic group]] (i.e., the [[semidirect product]] of a cyclic group and its [[automorphism group]]) is metabelian. This is because [[cyclic implies abelian automorphism group]] -- the automorphism group of a cyclic group is abelian. More generally, any group arising as the semidirect product of a cyclic group and any subgroup of its automorphism group is abelian.
+* Any [[dihedral group]] is metabelian: it has a cyclic normal subgroup and the quotient group is cyclic of order two. (This is a special case of the previous example). It turns out that dihedral groups are nilpotent only if their order is a power of two; further, the nilpotency class of a dihedral group of order <math>2^n</math> is <math>n - 1</math>.
+* Any [[generalized dihedral group]] is metabelian: it has an abelian normal subgroup and the quotient group is cyclic of order two.
-{{obtainedbyapplyingthe|meta operator|Abelian group}}
+===Particular examples===
-The property of being metabelian arises by applying the [[meta operator]] to the [[group property]] of being [[Abelian group|Abelian]]. Equivalently metabelian can be described as Abelian-by-Abelian, where ''by'' denotes the [[group extension operator]].
+* [[Symmetric group:S3|The symmetric group of degree three]], which is also the dihedral group of order six and degree three, is metabelian. It is the smallest example of a metabelian non-abelian group; it is also not nilpotent. The abelian normal subgroup in this case is the alternating group, i.e., the subgroup generated by a <math>3</math>-cycle <math>\langle (1,2,3) \rangle</math>, and the quotient group is cyclic of order two.
+* [[Dihedral group:D8|The dihedral group of order eight]] (degree four) is metabelian. We have several choices here of the abelian normal subgroup. We can take the center as the abelian normal subgroup, in which case the quotient is isomorphic to a [[Klein four-group]]. Alternatively, we could take as our abelian normal subgroup any of the subgroups of order four.
+* [[Alternating group:A4|the alternating group of degree four]] (order twelve) is metabelian. It has an abelian normal subgroup of order four: the Klein four-subgroup comprising the identity and the three double transpositions <math>(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)</math>. The quotient is a cyclic group of order three.
 ==Relation with other properties==
@@ Line 27: / Line 54: @@
 ===Stronger properties===
-* [[Abelian group]]
+{| class="sortable" border="1"
-* [[Group of nilpotence class two]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Center-is-commutator group]]
+|-
-* [[Metacyclic group]]
+| [[Weaker than::cyclic group]] || || || || {{intermediate notions short|metabelian group|cyclic group}}
+|-
+| [[Weaker than::abelian group]] || || || || {{intermediate notions short|metabelian group|abelian group}}
+|-
+| [[Weaker than::group of nilpotency class two]] || commutator subgroup is central || || || {{intermediate notions short|metabelian group|group of nilpotency class two}}
+|-
+| [[Weaker than::metacyclic group]] || cyclic normal subgroup, cyclic quotient || || || {{intermediate notions short|metabelian group|metacyclic group}}
+|-
+| [[Weaker than::group whose automorphism group is abelian]] || the [[automorphism group]] is an [[abelian group]] || || || {{intermediate notions short|metabelian group|group whose automorphism group is abelian}}
+|}
 ===Weaker properties===
-* [[Solvable group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::group with nilpotent derived subgroup]] || [[derived subgroup]] is nilpotent || || || {{intermediate notions short|group with nilpotent derived subgroup|metabelian group}}
+|-
+| [[Stronger than::abelian-by-nilpotent group]] || [[abelian normal subgroup]], nilpotent quotient || || || {{intermediate notions short|abelian-by-nilpotent group|metabelian group}}
+|-
+| [[Stronger than::solvable group]] || || || || {{intermediate notions short|solvable group|metabelian group}}
+|-
+| [[Stronger than::group satisfying subnormal join property]] || join of subnormal subgroups is subnormal || [[metabelian implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|metabelian group}}
+|-
+| [[Stronger than::metanilpotent group]] || nilpotent normal subgroup, nilpotent quotient || || || {{intermediate notions short|metanilpotent group|metabelian group}}
+|}
 ==Metaproperties==