# 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.

## Contents

## Statement of classification

### Abelian groups

There is a unique Hall-Senior *family*, called .

The nature and classification of the five *abelian* groups of order is the same for both the and odd 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 | GAP ID (2nd part) case | Hall-Senior symbol | Hall-Senior number |
---|---|---|---|---|---|

4 | cyclic group of prime-fourth order | cyclic group:Z16 | 1 | 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 | 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 | 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 | 2 | |

1 + 1 + 1 + 1 | elementary abelian group of prime-fourth order | elementary abelian group:E16 | 14 | 1 |

### Class two groups

There is a unique Hall-Senior family, called . This family is characterized as follows: the derived subgroup is isomorphic to cyclic group:Z2, the inner automorphism group is Klein four-group. If is the whole group, then the commutator map 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 | 6 | ||

direct product of Q8 and Z2 | 7 | ||

central product of D8 and Z4 | 8 | ||

SmallGroup(16,3) | 9 | ||

nontrivial semidirect product of Z4 and Z4 | 10 | ||

M16 | 11 |

### Class three groups

There is a unique Hall-Senior family, called . 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 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 | 12 | ||

semidihedral group:SD16 | 13 | ||

generalized quaternion group:Q16 | 14 |

## Facts used

- Prime power order implies not centerless
- Cyclic over central implies abelian
- Equivalence of definitions of group of prime order
- Lagrange's theorem
- Classification of groups of prime-square order
- 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 of order , nilpotency class two.

**To prove**: If denotes the center of , then is isomorphic to the Klein four-group, the derived subgroup is isomorphic to cyclic group:Z2, and the alternating bilinear map to is defined as follows: the image of a pair of unequal non-identity elements is the non-identity element of , and the image of equal elements and the image of a pair containing the identity element is the identity element.

**Proof**: Let .

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | is nontrivial. | Fact (1) | has order | Fact-direct | |

2 | cannot have order 16. | -- | is non-abelian | ||

3 | cannot have order 8. | Facts (2), (3), (4) | is non-abelian, has order 16. | [SHOW MORE] | |

4 | has order either 2 or 4. | Fact (4) | has class two. | Steps (1),(2),(3) | [SHOW MORE] |

5 | If has order 2, then has order 2. | is non-abelian and has class two | [SHOW MORE] | ||

6 | If has order 2, then we get a contradiction | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| |||

7 | If has order 4, then is isomorphic to a Klein four-group, has order two, and the commutator map 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

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## 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

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