# Every finite solvable group is a subgroup of a finite group having subgroups of all orders dividing the group order

This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.
View a complete list of embeddability theorems

## Statement

Let  be a finite solvable group. Then, there exists a finite group  that is a Group having subgroups of all orders dividing the group order (?), and containing a subgroup isomorphic to .

## Converse

The converse of the statement is true: if a group can be embedded as a subgroup of a finite group having subgroups of all orders dividing the group order, it must be a finite solvable group. This follows from the fact that having subgroups of all orders dividing the group order implies solvable (which in turn follows from Hall's theorem), combined with the fact that solvability is subgroup-closed.

## Facts used

1. ECD condition for pi-subgroups in solvable groups: This is an extended version of Sylow's theorem in finite solvable groups, stating that Hall subgroups of all permissible orders exist.
2. A cyclic group has subgroups of all orders dividing its order.

## Proof

Given: A finite solvable group  of order , with  prime and .

To prove: The direct product  has subgroups of all orders dividing its order, where . Note that this direct product contains , isomorphic to , so this is sufficient.

Proof:  has order:

.

Now, consider any divisor  of the order of , say:

.

We construct a subgroup of  of this order. First, define  as  if  and  if . Then, find a subgroup  of  of order:

.

Now, , so consider:

.

By fact (1), we conclude that  has a normal subgroup  of this order. Then,  is a subgroup of  of order , as required.