Odd-order and CN implies solvable

The result stated here is superseded by the following result, which is both stronger and simpler: odd-order implies solvable. In other words, the latter result has weaker and easier-to-verify hypotheses, and/or stronger and easier-to-use conclusions.
The main purpose of including this result is that it has a considerably easier proof, and/or was historically proved before the stronger result.

Statement

Any odd-order group that is also a CN-group (and hence a finite CN-group) must be a solvable group (and hence, a finite solvable group).

Note that the odd-order theorem says that every odd-order group is solvable, and therefore, this result is too weak to be of any use beyond what the odd-order theorem already tells us. However, it is a lot easier to prove than the odd-order theorem, and its utility lies in the way it helps pave the way for a proof of the odd-order theorem.

References

Journal references

• Solvability of groups of odd order by Walter Feit and John Griggs Thompson, Pacific Journal of Mathematics, Volume 13, Page 775 - 1029(Year 1963): This 255-page long paper gives a proof that odd-order implies solvable: any odd-order group (i.e., any finite group whose order is odd) is a solvable group.^{Project Euclid pageMore info}

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 409, Theorem 14.3.1, ^{More info}