Frattini-in-center odd-order p-group implies (mp plus 1)-power map is automorphism
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
Statement
Suppose is an odd prime, and
is a finite
-group (i.e., an Odd-order p-group (?): a Group of prime power order (?) with an odd prime). Then, if
is a Frattini-in-center group (?), i.e., if
, and
is any integer, then the map
is an automorphism of
.
Thus, it is a universal power automorphism. In the particular case that does not have exponent
, this gives a non-identity universal power automorphism.
Examples
For any odd prime , the smallest non-abelian examples are the groups of order
. There are two such examples: prime-cube order group:U(3,p) (GAP ID
) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (GAP ID
). The former has exponent
, so the
- power map is the identity automorphism. The latter has exponent
, so that
-power map is a non-identity universal power automorphism. In fact, this automorphism itself has order
.
For the case , these groups become prime-cube order group:U(3,3) and semidirect product of Z9 and Z3 respectively, both of order 27.
Related facts
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Square map is endomorphism iff abelian
- Inverse map is automorphism iff abelian
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)
- nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))
Facts used
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Abelian implies universal power map is endomorphism
- (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism
- kth power map is bijective iff k is relatively prime to the order
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
that is a Frattini-in-center group, i.e.,
is elementary abelian, or equivalently,
.
is an integer.
To prove: The map is an automorphism of
.
Proof:
Step no. | Assertion | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | Define ![]() ![]() ![]() ![]() |
Fact (1) | ![]() ![]() |
-- | Fact+Given direct. |
2 | The image of ![]() ![]() |
![]() |
[SHOW MORE] | ||
3 | The map ![]() ![]() ![]() |
Fact (2) | Steps (1), (2) | [SHOW MORE] | |
4 | The map ![]() ![]() |
Fact (3) | Step (3) | Fact+Step direct, setting ![]() | |
5 | The map ![]() ![]() ![]() |
Fact (4) | [SHOW MORE] | ||
6 | The map ![]() ![]() |
Steps (4), (5) | Step-combination-direct. |