Hall-Senior classification of groups of order 16

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.

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.

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:

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:

Facts used

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

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 $$Z = Z(P)$$.