# M16

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## Contents

## Definition

The group, sometimes denoted or , is defined as follows:

.

Here, denotes the identity element.

## Position in classifications

Get more information about groups of the same order at Groups of order 16#The list

Type of classification | Position/number in classification |
---|---|

GAP ID | , i.e., among groups of order 16 |

Hall-Senior number | 12 among groups of order 16 |

Hall-Senior symbol |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

### Important properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No | do not commute | |

metacyclic group | Yes | has cyclic subgroup of order eight, quotient group isomorphic to is cyclic of order two. | |

metabelian group | Yes | follows from being metacyclic. |

### Other properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

finite group that is 1-isomorphic to an abelian group | Yes | via cocycle halving generalization of Baer correspondence | See element structure of groups of order 16#1-isomorphism |

Schur-trivial group | Yes | ||

directly indecomposable group | Yes | cannot be expressed as a direct product of proper subgroups; all proper subgroups are abelian, but the group is non-abelian. | |

splitting-simple group | No | it is an internal semidirect product of (a normal subgroup) and . | |

centrally indecomposable group | Yes | it cannot be expressed as a central product of strictly smaller subgroups. | |

group of nilpotency class two | Yes | ||

group in which every normal subgroup is characteristic | No | is normal but not characteristic. | |

ambivalent group | No | are not conjugate. | This also means that it is not a rational group or rational-representation group. |

## Elements

`Further information: element structure of M16`

### Conjugacy class structure

Conjugacy class | Size of conjugacy class | Order of elements in conjugacy class | Centralizer of first element of class |
---|---|---|---|

1 | 1 | whole group | |

1 | 2 | whole group | |

1 | 4 | whole group | |

1 | 4 | whole group | |

2 | 8 | ||

2 | 8 | ||

2 | 2 | ||

2 | 4 | ||

2 | 8 | ||

2 | 8 |

### Automorphism class structure

Equivalence class under automorphisms | Size of equivalence class | Number of conjugacy classes in it | Size of each conjugacy class | Characterization(s) up to 1-isomorphism | Characterization(s) involving commutation relationships |
---|---|---|---|---|---|

1 | 1 | 1 | identity element | identity element | |

1 | 1 | 1 | unique non-identity element that is a fourth power | unique non-identity element that is a commutator | |

2 | 2 | 1 | squares that are not fourth powers | elements in the center but not in the derived subgroup | |

8 | 4 | 2 | elements of order eight | ||

2 | 1 | 2 | elements of order two that are not squares | ||

2 | 1 | 2 | elements of order four that are not squares |

### 1-isomorphism

The group is 1-isomorphic to the group direct product of Z8 and Z2. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the cocycle halving generalization of Baer correspondence, where the intermediary is a class two Lie cring.

## Subgroups

`Further information: subgroup structure of M16`

To describe subgroups, we use the defining presentation given at the beginning:

.

Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes(=1 iff automorph-conjugate subgroup) | Size of each conjugacy class(=1 iff normal subgroup) | Total number of subgroups(=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Subnormal depth | Nilpotency class |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 16 | 1 | 1 | 1 | M16 | 1 | 0 | |

derived subgroup of M16 | cyclic group:Z2 | 2 | 8 | 1 | 1 | 1 | direct product of Z4 and Z2 | 1 | 1 | |

other subgroups of order two | , | cyclic group:Z2 | 2 | 8 | 1 | 2 | 2 | -- | 2 | 1 |

center of M16 | cyclic group:Z4 | 4 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 1 | |

other cyclic subgroup of order 4 | cyclic group:Z4 | 4 | 4 | 1 | 1 | 1 | cyclic group:Z4 | 1 | 1 | |

V4 in M16 | Klein four-group | 4 | 4 | 1 | 1 | 1 | cyclic group:Z4 | 1 | 1 | |

Z8 in M16 | |
cyclic group:Z8 | 8 | 2 | 2 | 1 | 2 | cyclic group:Z2 | 1 | 1 |

direct product of Z4 and Z2 in M16 | direct product of Z4 and Z2 | 8 | 2 | 1 | 1 | 1 | cyclic group:Z2 | 1 | 1 | |

whole group | all elements | M16 | 16 | 1 | 1 | 1 | 1 | trivial group | 1 | 1 |

Total (9 rows) | -- | -- | -- | -- | 10 | -- | 11 | -- | -- | -- |

## Distinguishing features

### Smallest of its kind

- This is a minimum order example of a non-abelian finite group that is 1-isomorphic to an abelian group -- it is 1-isomorphic to direct product of Z8 and Z2. It is, however, not the only such example: the other example is central product of D8 and Z4. See element structure of groups of order 16#1-isomorphism for more details.
- This is a minimum order example of a nilpotent group that is not a UL-equivalent group, i.e., the upper central series and lower central series are not the same. However, it is not the only example. In fact, all groups of order 16 and class two share this property with it. The other examples are SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2, and central product of D8 and Z4.
- This is a minimum order example of a situation where a group has two characteristic subgroups that are both isomorphic to each other but are distinct. Both these are cyclic subgroups of order four. It is, however, not the only example of this order; there are other examples where a similar behavior occurs, albeit for different orders of subgroups -- for instance, order 2 (in nontrivial semidirect product of Z4 and Z4).

## GAP implementation

### Group ID

This finite group has order 16 and has ID 6 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,6)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(16,6);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [16,6]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Hall-Senior number

This group of prime power order has order 16 and has Hall-Senior number 11 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(16,11)`

WARNING: There is some disagreement between the GAP 3 catalogue numbers and the Hall-Senior numbers for some abelian groups, but it does not affect this group.

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := Gap3CatalogueGroup(16,11);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's Gap3CatalogueIdGroup function:

`Gap3CatalogueIdGroup(G) = [16,11]`

or just do:

`Gap3CatalogueIdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Description by presentation

The group can be defined using a presentation as follows:

gap> F := FreeGroup(2);; gap> G := F/[F.1^8,F.2^2,F.2*F.1*F.2*F.1^(-5)]; <fp group on the generators [ f1, f2 ]> gap> IdGroup(G); [ 16, 6 ]