View source for Supersolvable group

[[importance rank::2| ]]
==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 group|cyclic]].

===Definition with symbols===

A group <math>G</math> is said to be '''supersolvable''' if there exists a normal series:

<math>1 = H_0 \le H_1 \le H_2 \le \ldots \le H_n = G</math>

where each <math>H_i \triangleleft G</math> and further, each <math>H_{i+1}/H_i</math> is [[cyclic group|cyclic]].

==Examples==

{{group property see examples}}

==Metaproperties==

{| 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.
|}


{{semibasicdef}}
{{pivotal group property}}
{{variationof|solvability}}
{{finite-at|finite supersolvable group}}

==Relation with other properties==

===Conjunction with other properties===

{| class="sortable" border="1"
! Conjunction !! Other component of conjunction
|-
| [[finite supersolvable group]] || [[finite group]]
|}
===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::finite abelian group]] || || || || {{intermediate notions short|supersolvable group|finite abelian group}}
|-
| [[Weaker than::finitely generated abelian group]] || || || || {{intermediate notions short|supersolvable group|finitely generated abelian group}}
|-
| [[Weaker than::finite nilpotent group]] || || || || {{intermediate notions short|supersolvable group|finite nilpotent group}}
|-
| [[Weaker than::finitely generated nilpotent group]] || || || || {{intermediate notions short|supersolvable group|finitely generated nilpotent group}}
|-
| [[Weaker than::finite supersolvable group]] || || || || {{intermediate notions short|supersolvable group|finite supersolvable group}}
|}
===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::polycyclic group]] || has a [[subnormal series]] with cyclic quotients || || || {{intermediate notions short|polycyclic group|supersolvable group}}
|-
| [[Stronger than::solvable group]] || || || || {{intermediate notions short|solvable group|supersolvable group}}
|-
| [[Stronger than::finitely generated solvable group]] || || || || {{intermediate notions short|finitely generated solvable group|supersolvable group}}
|-
| [[Stronger than::group with nilpotent derived subgroup]] || ||[[Supersolvable implies nilpotent commutator subgroup]] || ||
|}

==Study of the notion==

{{msc class|20F16}}