Frattini-in-center odd-order p-group implies p-power map is endomorphism
This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
View other such facts for p-groups|View other such facts for finite groups
Contents
Statement
Suppose is an odd prime, and is a finite -group (i.e., a group of prime power order) that is a Frattini-in-center group: the Frattini subgroup of is contained in its center. Then, the map is an endomorphism of .
Note that this makes it a universal power endomorphism, i.e., an endomorphism described everywhere as raising to a certain power. The endomorphism is nontrivial only if does not itself have exponent .
Examples
The smallest non-abelian examples for any odd prime are the two non-abelian groups of order , namely unitriangular matrix group:UT(3,p) (GAP ID ) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (GAP ID ). Of these two groups, the former has exponent , so the -power map is the trivial endomorphism. The latter has exponent , so the -power map is a nontrivial endomorphism.
In the case , these groups are unitriangular matrix group:UT(3,3) and semidirect product of Z9 and Z3 respectively. Both groups have order .
Related facts
Failure at the prime two
- Square map is endomorphism iff abelian, combined with the fact that there exist non-Abelian 2-groups that are Frattini-in-center.
Facts with similar proofs
Related facts about power maps
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)
- Inverse map is automorphism iff abelian
- Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism
Facts used
- Frattini-in-center p-group implies derived subgroup is elementary abelian
- Formula for powers of product in group of class two
Proof
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
Given: An odd prime . A finite -group , such that is elementary Abelian.
To prove: The map is an endomorphism of . Specifically for any .
Proof:
Step no. | Assertion | Given data used | Facts used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | The derived subgroup is elementary abelian. In particular, is the identity element for any . | is Frattini-in-center | Fact (1) | -- | Follows directly from fact (1). |
2 | divides . | is an odd prime. | -- | -- | Basic properties of divisibility. Note that this breaks down for , because of the in the denominator. |
3 | is the identity element for all . | Steps (1), (2) | By step (1), is the identity element, so the order of divides . Since divides , the order of divides , so is the identity element. | ||
4 | We have the formula for all . | is Frattini-in-center, and hence class two. | Fact (2) | -- | Because is Frattini-in-center, the quotient by the center is elementary abelian, and hence abelian, so has class at most two. Thus, we can use fact (2) to get the formula. |
5 | for all | Steps (3), (4) | This follows directly by plugging in the conclusion of step (3) into step (4). |
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 183-184, Lemma 3.9, Section 5.3 (-automorphisms of -groups)