# Group having subgroups of all orders dividing the group order

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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

### 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 $q = p^r, r > 1$, the general affine group $GA(1,q)$ 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 $G$ 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