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 .

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.