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.

Classification of abelian groups

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 $p=2$
4	cyclic group of prime-fourth order	cyclic group:Z16	1
3 + 1	direct product of cyclic group of prime-cube order and cyclic group of prime order	direct product of Z8 and Z2	5
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 + 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
1 + 1 + 1 + 1	elementary abelian group of prime-fourth order	elementary abelian group:E16	14