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
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 .
- 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.
- A cyclic group has subgroups of all orders dividing its order.
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.