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

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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 .

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.