Burnside's theorem on coprime automorphisms and Frattini subgroup

Statement

Suppose $P$ is a finite $p$-group, i.e., a group of prime power order. Suppose $\varphi$ is an automorphism of $P$ whose order (as an element of the automorphism group of $P$) is relatively prime to $P$. Then, if $\varphi$ induces the identity on $P/\Phi (P)$, $\varphi$ is the identity automorphism of $P$. Here $\Phi (P)$ is the Frattini subgroup of $P$.

Equivalently the kernel of the map:

$\operatorname {Aut} (P)\to \operatorname {Aut} (P/\Phi (P))$

is a $p$-group. We also say that the Frattini subgroup of a $p$-group is a quotient-coprime automorphism-faithful subgroup.

Related facts

Generalizations

• Frattini subgroup of finite group is quotient-coprime automorphism-faithful: This generalizes the statement of Burnside's theorem to an arbitrary finite group.

Similar facts

• Stability group of subnormal series of p-group is p-group: The result is similar, and it uses a similar proof technique.
• Omega-1 of odd-order p-group is coprime automorphism-faithful

Facts used

1. Frattini subgroup of finite group is quotient-coprime automorphism-faithful

Proof

Given: A finite $p$-group $P$, an automorphism $\varphi$ of $P$ whose order is relatively prime to $p$. Further $\varphi$ induces the identity automorphism on $P/\Phi (P)$.

To prove: $\varphi$ is the identity automorphism.

Proof: The proof follows directly from fact (1), which is actually the general formulation for finite groups.

(NOTE: We can use the Burnside's basis theorem instead; instead of picking a coset representative for every coset, we pick a Burnside basis, and then find a coset representative for each coset in the Burnside basis, that is fixed under $\sigma$.)

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 199, Theorem 1.4 (Chapter 5)
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 199, Exercise 26(e) and (f) (Section 6.1)