Supersolvable group: Difference between revisions

Revision as of 02:19, 19 June 2011

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 that is pivotal (i.e., important) among existing group properties
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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

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.

Relation with other properties

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 commutator subgroup		Supersolvable implies nilpotent commutator subgroup

Examples

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

Facts

For a complete list of facts about supersolvable groups:

Special:SearchByProperty/Fact-20about/Supersolvable-20group

Commutator subgroup is nilpotent

It turns out that for any supersolvable group, the commutator subgroup is nilpotent.

Normal abelian subgroup properly containing the center

In a supersolvable group, there is an abelian normal subgroup properly containing the center. This fact turns out to be crucially important for proving that every supersolvable group is a monomial-representation group.

Every representation is monomial

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

Metaproperty name	Satisfied?	Proof	Statement with symbols
quasivarietal group property	Yes	supersolvability is quasivarietal	closed under taking subgroups, quotients, and finite direct products
subgroup-closed group property	Yes	supersolvability is subgroup-closed	If $G$ is supersolvable, and $H \leq G$ is a subgroup, $H$ is a subgroup.
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.

Study of the notion

Mathematical subject classification

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

@@ Line 73: / Line 73: @@
 ==Metaproperties==
-{{S-closed}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-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.
+|-
+| [[satisfies metaproperty::quasivarietal group property]] || Yes || [[supersolvability is quasivarietal]] || closed under taking subgroups, quotients, and finite direct products
-{{Q-closed}}
+|-
+| [[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 a subgroup.
-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.
+|-
+| [[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.
-{{P-closed}}
+|-
+| [[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.
-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}}