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

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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>. | ||

+ | |||

+ | ==Examples== | ||

+ | |||

+ | ===Extreme examples=== | ||

+ | |||

+ | * The [[trivial group]] and any [[group of prime order]] are obvious examples where this holds. | ||

+ | |||

+ | ===Other examples=== | ||

+ | |||

+ | * Any [[group of prime power order]] satisfies this. | ||

+ | * Any [[finite abelian group]] and more generally any [[finite nilpotent group]] or even any [[finite supersolvable group]] satisfies this. | ||

+ | * [[Symmetric group:S4]] satisfies this condition, though it is not supersolvable. | ||

+ | |||

+ | ===Non-examples=== | ||

+ | |||

+ | * 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. | ||

+ | * [[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. | ||

+ | |||

+ | ==Metaproperties== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||

+ | |- | ||

+ | | [[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. | ||

+ | |- | ||

+ | | [[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. | ||

+ | |- | ||

+ | | [[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. | ||

+ | |} | ||

==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

## Contents

## Definition

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

## Examples

### Extreme examples

- The trivial group and any group of prime order are obvious examples where this holds.

### Other examples

- Any group of prime power order satisfies this.
- Any finite abelian group and more generally any finite nilpotent group or even any finite supersolvable group satisfies this.
- Symmetric group:S4 satisfies this condition, though it is not supersolvable.

### Non-examples

- 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.
- Alternating group:A4 does not satisfy this condition. More generally, for any prime power , the general affine group does not satisfy this condition.

## 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 satisfying this condition and a subgroup of 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 satisfying this condition and a normal subgroup of such that the quotient group 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 and are finite groups both satisfying the property, then the external direct product 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

- Group having a Sylow tower:
`For full proof, refer: Subgroups of all orders dividing the group order not implies Sylow tower, Sylow tower not implies subgroups of all orders dividing the group order`

## Facts

- Every finite solvable group can be embedded inside a finite group having subgroups of all orders dividing the group order.
`For full proof, refer: Every finite solvable group is a subgroup of a finite group having subgroups of all orders dividing the group order`