Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Statement 2 Related facts 2.1 Generalizations 2.2 Similar facts 3 Facts used 4 Proof 5 References 5.1 Textbook references

Statement

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

Equivalently the kernel of the map:

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 of P whose order is relatively prime to p. Further induces the identity automorphism on P / Φ(P).

To prove: 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 σ.)

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)