# Group having subgroups of all orders dividing the group order

From Groupprops

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 orders 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`