Group having subgroups of all orders dividing the group order

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

• 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 $q=p^{r},r>1$, 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			|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