Supersolvable group: Difference between revisions

Latest revision as of 17:17, 26 July 2013

Definition

Symbol-free definition

A group is said to be supersolvable if it has a normal series (wherein all the members are normal in the whole group) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.

Definition with symbols

A group $G$ is said to be supersolvable if there exists a normal series:

$1 = H_{0} \leq H_{1} \leq H_{2} \leq \dots \leq H_{n} = G$

where each $H_{i} ◃ G$ and further, each $H_{i + 1} / H_{i}$ is cyclic.

Examples

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
pseudovarietal group property	Yes	supersolvability is pseudovarietal	Supersolvability is closed under taking subgroups, quotients, and finite direct products (see below).
subgroup-closed group property	Yes	supersolvability is subgroup-closed	If $G$ is supersolvable, and $H \leq G$ is a subgroup, $H$ is supersolvable.
quotient-closed group property	Yes	supersolvability is quotient-closed	If $G$ is supersolvable, and $H$ is a normal subgroup of $G$ , the quotient group $G / H$ is supersolvable.
finite direct product-closed group property	Yes	supersolvability is finite direct product-closed	If $G_{1}, G_{2}, \dots, G_{n}$ are all supersolvable, the external direct product $G_{1} \times G_{2} \times \dots \times G_{n}$ is also supersolvable.

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
VIEW: Definitions built on this | Facts about this: (facts closely related to Supersolvable group, all facts related to Supersolvable group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

This is a variation of solvability|Find other variations of solvability |

The version of this for finite groups is at: finite supersolvable group

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction
finite supersolvable group	finite group

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group				\|FULL LIST, MORE INFO
finitely generated abelian group				\|FULL LIST, MORE INFO
finite nilpotent group				\|FULL LIST, MORE INFO
finitely generated nilpotent group				\|FULL LIST, MORE INFO
finite supersolvable group				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
polycyclic group	has a subnormal series with cyclic quotients		\|FULL LIST, MORE INFO
solvable group			\|FULL LIST, MORE INFO
finitely generated solvable group			\|FULL LIST, MORE INFO
group with nilpotent derived subgroup		Supersolvable implies nilpotent commutator subgroup

Study of the notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F16

@@ Line 1: / Line 1: @@
-{{semibasicdef}}
+[[importance rank::2| ]]
-{{pivotal group property}}
-{{variationof|solvability}}
-{{finite-at|finite supersolvable group}}
 ==Definition==
@@ Line 17: / Line 14: @@
 where each <math>H_i \triangleleft G</math> and further, each <math>H_{i+1}/H_i</math> is [[cyclic group|cyclic]].
-==Relation with other properties==
+==Examples==
-===Stronger properties===
+{{group property see examples}}
-* [[Nilpotent group]] (for [[finite group]]s)
+==Metaproperties==
-===Weaker properties===
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::pseudovarietal group property]] || Yes || [[supersolvability is pseudovarietal]] || Supersolvability is closed under taking subgroups, quotients, and finite direct products (see below).
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[supersolvability is subgroup-closed]] || If <math>G</math> is supersolvable, and <math>H \le G</math> is a subgroup, <math>H</math> is supersolvable.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[supersolvability is quotient-closed]] || If <math>G</math> is supersolvable, and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is supersolvable.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[supersolvability is finite direct product-closed]] || If <math>G_1,G_2,\dots,G_n</math> are all supersolvable, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also supersolvable.
+|}
-* [[Polycyclic group]]
-* [[Solvable group]]
-* [[Monomial-representation group]]
-* [[Nilpotent-commutator group]]
-==Facts==
+{{semibasicdef}}
+{{pivotal group property}}
+{{variationof|solvability}}
+{{finite-at|finite supersolvable group}}
-For a complete list of facts about supersolvable groups:
+==Relation with other properties==
-[[Special:SearchByProperty/Fact-20about/Supersolvable-20group]]
+===Conjunction with other properties===
-For specific kinds of facts:
+{| class="sortable" border="1"
+! Conjunction !! Other component of conjunction
+|-
+| [[finite supersolvable group]] || [[finite group]]
+|}
+===Stronger properties===
-* [[:Category:Subgroup property implications in supersolvable groups]]
+{| class="sortable" border="1"
-* [[:Category:Subgroup property non-implications in supersolvable groups]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[:Category:Subgroup metaproperty satisfactions in supersolvable groups]]
+|-
-* [[:Category:Subgroup metaproperty dissatisfactions in supersolvable groups]]
+| [[Weaker than::finite abelian group]] || || || || {{intermediate notions short|supersolvable group|finite abelian group}}
+|-
-===Derived subgroup is nilpotent===
+| [[Weaker than::finitely generated abelian group]] || || || || {{intermediate notions short|supersolvable group|finitely generated abelian group}}
+|-
-It turns out that for any supersolvable group, the commutator subgroup is nilpotent.
+| [[Weaker than::finite nilpotent group]] || || || || {{intermediate notions short|supersolvable group|finite nilpotent group}}
+|-
-===Normal Abelian subgroup properly containing the center===
+| [[Weaker than::finitely generated nilpotent group]] || || || || {{intermediate notions short|supersolvable group|finitely generated nilpotent group}}
+|-
-In a supersolvable group, there is a normal Abelian group properly containing the center. This fact turns out to be crucially important for proving that every supersolvable group is a monomial-representation group.
+| [[Weaker than::finite supersolvable group]] || || || || {{intermediate notions short|supersolvable group|finite supersolvable group}}
+|}
-===Every representation is monomial===
+===Weaker properties===
-It turns out that every representation of a supersolvable group is monomial. In other words, every irreducible representation of a supersolvable group can be induced from a one-dimensional representation of some subgroup.
-===Elements of odd order form a characteristic subgroup===
-==Metaproperties==
-{{S-closed}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-Any subgroup of a supersolvable group is supersolvable. The normal series for the subgroup can be obtained simply by intersecting the normal series of the group, with the subgroup.
+|-
+| [[Stronger than::polycyclic group]] || has a [[subnormal series]] with cyclic quotients || || || {{intermediate notions short|polycyclic group|supersolvable group}}
-{{Q-closed}}
+|-
+| [[Stronger than::solvable group]] || || || || {{intermediate notions short|solvable group|supersolvable group}}
-Any quotient of a supersolvable group is supersolvable. The normal series for the quotient is obtained by taking the image of the normal series for the original group via the quotient map.
+|-
+| [[Stronger than::finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|supersolvable group}}
-{{P-closed}}
+|-
+| [[Stronger than::group with nilpotent derived subgroup]] || ||[[Supersolvable implies nilpotent commutator subgroup]] || ||
-Any direct product of supersolvable groups is supersolvable. In fact, more generally, any central product of supersolvable groups is supersolvable.
+|}
 ==Study of the notion==
 {{msc class|20F16}}
-==External links==
-===Definition links===
-* {{wp|Supersolvable group}}
-* {{planetmath|SupersolvableGroup}}