Hall-Senior classification of groups of order 16

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This article describes the classification of the groups of order 16 by using the ideas of Hall-Senior genus and Hall-Senior family. This is one of many mutually similar classification approaches.

Statement of classification

Abelian groups

There is a unique Hall-Senior family, called \Gamma_1.

The nature and classification of the five abelian groups of order p^4 is the same for both the p = 2 and odd p cases; the abelian groups are classified by the set of unordered integer partitions of the number 4. This follows from the structure theorem for finitely generated abelian groups. We do not discuss the classification of abelian groups further in this article, since it is common across all classification approaches.

Partition of 4 Corresponding abelian group (in general) Corresponding abelian group case p = 2 GAP ID (2nd part) case p = 2 Hall-Senior symbol Hall-Senior number
4 cyclic group of prime-fourth order cyclic group:Z16 1 (4) 5
3 + 1 direct product of cyclic group of prime-cube order and cyclic group of prime order direct product of Z8 and Z2 5 (31) 4
2 + 2 direct product of cyclic group of prime-square order and cyclic group of prime-square order direct product of Z4 and Z4 2 (2^2) 3
2 + 1 + 1 direct product of cyclic group of prime-square order and elementary abelian group of prime-square order direct product of Z4 and V4 10 (21^2) 2
1 + 1 + 1 + 1 elementary abelian group of prime-fourth order elementary abelian group:E16 14 (1^4) 1

Class two groups

There is a unique Hall-Senior family, called \Gamma_2. This family is characterized as follows: the derived subgroup is isomorphic to cyclic group:Z2, the inner automorphism group is Klein four-group. If P is the whole group, then the commutator map P/Z(P) \times P/Z(P) \to [P,P] is fixed as the unique alternating bilinear map possible.

Here is a classification of the groups in this family:

Group Hall-Senior symbol Hall-Senior number Hall-Senior genus
direct product of D8 and Z2 16\Gamma_2a_1 6 16\Gamma_2a
direct product of Q8 and Z2 16\Gamma_2a_2 7 16\Gamma_2a
central product of D8 and Z4 16\Gamma_2b 8 16\Gamma_2b
SmallGroup(16,3) 16\Gamma_2c_1 9 16\Gamma_2c
nontrivial semidirect product of Z4 and Z4 16\Gamma_2c_2 10 16\Gamma_2c
M16 16\Gamma_2d 11 16\Gamma_2d

Class three groups

There is a unique Hall-Senior family, called \Gamma_3. This family is characterized as follows: the derived subgroup is isomorphic to cyclic group:Z4, the inner automorphism group is dihedral group:D8, and the commutator map P/Z(P) \times P/Z(P) \to [P,P] is fixed as the unique alternating bilinear map possible.

In fact, there is a unique Hall-Senior genus.

Here is a classification of the groups in this family:

Group Hall-Senior symbol Hall-Senior number Hall-Senior genus
dihedral group:D16 16\Gamma_3a_1 12 16\Gamma_3a
semidihedral group:SD16 16\Gamma_3a_2 13 16\Gamma_3a
generalized quaternion group:Q16 16\Gamma_3a_3 14 16\Gamma_3a

Facts used

  1. Prime power order implies not centerless
  2. Cyclic over central implies abelian
  3. Equivalence of definitions of group of prime order
  4. Lagrange's theorem
  5. Classification of groups of prime-square order
  6. Class two implies commutator map is endomorphism

Proof of classification for nilpotency class two

Proof of the uniqueness of the Hall-Senior family

'Given: A non-abelian group P of order 16, nilpotency class two.

To prove: If Z(P) denotes the center of P, then P/Z(P) is isomorphic to the Klein four-group, the derived subgroup [P,P] is isomorphic to cyclic group:Z2, and the alternating bilinear map P/Z(P) \times P/Z(P) to [P,P] is defined as follows: the image of a pair of unequal non-identity elements is the non-identity element of [P,P], and the image of equal elements and the image of a pair containing the identity element is the identity element.

Proof: Let Z = Z(P).

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 Z is nontrivial. Fact (1) P has order 2^4 Fact-direct
2 Z cannot have order 16. -- P is non-abelian
3 Z cannot have order 8. Facts (2), (3), (4) P is non-abelian, has order 16. [SHOW MORE]
4 Z has order either 2 or 4. Fact (4) P has class two. Steps (1),(2),(3) [SHOW MORE]
5 If Z has order 2, then [P,P] has order 2. P is non-abelian and has class two [SHOW MORE]
6 If Z has order 2, then we get a contradiction PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
7 If Z has order 4, then P/Z is isomorphic to a Klein four-group, [P,P] has order two, and the commutator map P/Z \times P/Z \to [P,P] is as described. Facts (1), (4), (5), (6) [SHOW MORE]

Construction of possible values of Hall-Senior genus

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Classification of the groups

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Proof of classification for nilpotency class three

Proof of the uniqueness of the Hall-Senior family

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Construction of possible values of Hall-Senior genus

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Classification of the groups