Difference between revisions of "Group having subgroups of all orders dividing the group order"

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(Weaker properties)
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Suppose <math>G</math> is a [[finite group]] of order <math>n</math>. We say that <math>G</math> is a '''group having subgroups of all orders dividing the group order''' if, for any positive divisor <math>d</math> of <math>n</math>, there exists a subgroup <math>H</math> of <math>G</math> of order <math>d</math>.
 
Suppose <math>G</math> is a [[finite group]] of order <math>n</math>. We say that <math>G</math> is a '''group having subgroups of all orders dividing the group order''' if, for any positive divisor <math>d</math> of <math>n</math>, there exists a subgroup <math>H</math> of <math>G</math> of order <math>d</math>.
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==Examples==
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===Extreme examples===
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* The [[trivial group]] and any [[group of prime order]] are obvious examples where this holds.
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===Other examples===
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* Any [[group of prime power order]] satisfies this.
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* Any [[finite abelian group]] and more generally any [[finite nilpotent group]] or even any [[finite supersolvable group]] satisfies this.
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* [[Symmetric group:S4]] satisfies this condition, though it is not supersolvable.
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===Non-examples===
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* By [Hall's theorem]], any such group must be a [[finite solvable group]]. Therefore any finite non-solvable group, such as [[alternating group:A5]], gives a counterexample.
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* [[Alternating group:A4]] does not satisfy this condition. More generally, for any [[prime power]] <math>q = p^r, r > 1</math>, the [[general affine group of degree one over a finite field|general affine group]] <math>GA(1,q)</math> does not satisfy this condition.
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==Metaproperties==
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{| class="sortable" border="1"
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
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|-
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| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[Having subgroups of all order dividing the group order is not subgroup-closed]] || It is possible to have a finite group <math>G</math> satisfying this condition and a subgroup <math>H</math> of <math>G</math> that does not satisfy this condition.
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|-
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| [[dissatisfies metaproperty::quotient-closed group property]] || No || [[Having subgroups of all orders dividing the group order is not quotient-closed]] || It is possible to have a finite group <math>G</math> satisfying this condition and a [[normal subgroup]] <math>H</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> does not satisfy the condition.
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|-
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| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[Having subgroups of all orders dividing the group order is finite direct product-closed]] || If <math>G_1</math> and <math>G_2</math> are finite groups both satisfying the property, then the [[external direct product]] <math>G_1 \times G_2</math> also satisfies the property.
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|}
  
 
==Relation with other properties==
 
==Relation with other properties==

Revision as of 02:26, 26 December 2015

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

Definition

Suppose G is a finite group of order n. We say that G is a group having subgroups of all orders dividing the group order if, for any positive divisor d of n, there exists a subgroup H of G of order d.

Examples

Extreme examples

Other examples

Non-examples

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No Having subgroups of all order dividing the group order is not subgroup-closed It is possible to have a finite group G satisfying this condition and a subgroup H of G that does not satisfy this condition.
quotient-closed group property No Having subgroups of all orders dividing the group order is not quotient-closed It is possible to have a finite group G satisfying this condition and a normal subgroup H of G such that the quotient group G/H does not satisfy the condition.
finite direct product-closed group property Yes Having subgroups of all orders dividing the group order is finite direct product-closed If G_1 and G_2 are finite groups both satisfying the property, then the external direct product G_1 \times G_2 also satisfies the property.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group of prime power order the order is a prime power prime power order implies subgroups of all orders dividing the group order (all the other examples, such as finite abelian groups of other orders) |FULL LIST, MORE INFO
finite abelian group finite and abelian: any two elements commute |FULL LIST, MORE INFO
finite nilpotent group finite and nilpotent; also equivalent to saying that there is a normal subgroup of every order dividing the group order Finite supersolvable group|FULL LIST, MORE INFO
finite supersolvable group finite and supersolvable; it has a normal series where all the quotient groups are cyclic groups finite supersolvable implies subgroups of all orders dividing the group order subgroups of all orders dividing the group order not implies supersolvable |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite solvable group finite and a solvable group Hall's theorem finite solvable not implies subgroups of all orders dividing the group order |FULL LIST, MORE INFO

Incomparable properties

Facts