Every finite group is the Fitting quotient of a p-dominated group for any prime p not dividing its order

History

This result was proved in a paper by Hartley and Robinson.

Statement

Suppose $H$ is a finite group and $p$ is a prime not dividing the order of $H$. Then, there exists a P-dominated group (?) whose quotient by its Fitting subgroup is $H$.

In other words, there is a Finite complete group (?) $G$ (i.e., a finite group that is complete: it is centerless and every automorphism is inner) such that the Fitting subgroup (?) $F(G)$ is a group of prime power order for the prime $p$, and the quotient group $G/F(G)$ is isomorphic to $H$. In fact, $G$ is the semidirect product of $F(G)$ and $H$.

Facts used

1. Bryant-Kovacs theorem

References

Journal references

• On finite complete groups by Brian Hartley and Derek John Scott Robinson, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 35,Number 1, Page 67 - 74(Year 1980): ^{PDF copy (Springerlink)More info}