Structure lemma for p-group with coprime automorphism group having automorphism trivial on invariant subgroups
Contents
Statement
Suppose is a Group of prime power order (?), and is a subgroup of , of order relatively prime to . Suppose there exists a non-identity element such that acts as the identity on every proper -invariant subgroup of . Then, we have the following conclusions about the structure of :
- . In other words, the commutator subgroup is contained in the center, and the quotient by the commutator subgroup is elementary Abelian.
- Either of two cases can occur: , in which case is a Special group (?), or is trivial, in which case is an Elementary Abelian group (?).
We also have the following conclusions about the action of on :
- acts nontrivially on , and the action of on is irreducible (in other words, there are no proper nontrivial -invariant subgroups).
- In case is a special group, also acts trivially on .
Facts used
- Stability group of subnormal series of p-group is p-group
- Omega-1 of Abelian p-group is coprime automorphism-faithful
- Maschke's averaging lemma
Proof
Given: A -group , a subgroup of such that has order relatively prime to . A non-identity element such that acts as the identity on every proper -invariant subgroup.
To prove: acts nontrivially on and trivially on . is elementary Abelian and acts irreducibly on . is either elementary Abelian or special.
Proof that acts nontrivially on
Since is a characteristic subgroup of , it is in particular -invariant. Thus, acts trivially on . If acted trivially on , fact (1) would yield that is the identity map, a contradiction. Thus, acts nontrivially on . In particular, acts nontrivially on .
Proof that is elementary Abelian and that the action of is irreducible
Let . Suppose the action of on is decomposable, so , where both and are proper -invariant. Their inverse images in are proper -invariant subgroups , and acts trivially on both and . Thus, acts trivially on , a contradiction.
Thus, the action of on is indecomposable.
Next, consider the induced action of on . Suppose is a proper subgroup of . Then, the inverse image of this in is a proper -invariant subgroup, so acts trivially on it, forcing to act trivially on . By fact (2), we see that this forces to be the identity on , a contradiction.
Thus, , so is elementary Abelian. The action of on is indecomposable, so by fact (3), it is irreducible.
Proof that is elementary Abelian or special
We first show that centralizes . PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Thus, . So, is an -invariant subgroup containing . Thus, is an -invariant subgroup of . By the irreducibility of the action of on , either is trivial or . Thus, either (in which case is trivial and is elementary Abelian) or (in which case , and so is a special group).
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Theorem 3.7, Page 181-183, Section 5.3 (-automorphisms of -groups)