# Generalized quaternion group:Q16

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## Contents

## Definition

The group , sometimes termed the **generalized quaternion group** of order , is a generalized quaternion group. It can be described by the following presentation:

.

Note that from these relations, and . This in turn forces that , forcing to have order two. We shall denote this element of order two, which is clearly central, as .

We can thus use an alternative presentation that requires only two generators:

## 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

### Arithmetic functions of a counting nature

## 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 | don't commute. | |

metacyclic group | Yes | is cyclic of order eight, quotient group is cyclic of order two. | |

metabelian group | Yes | follows from being metacyclic. |

### Other properties

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

finite group with periodic cohomology | finite group in which every abelian subgroup is cyclic | Yes | ||

Schur-trivial group | the Schur multiplier is trivial | Yes | follows from having periodic cohomology | |

maximal class group | finite p-group of class more than one whose class is one less than the prime-base logarithm of order | Yes | class is 3, prime-base logarithm of order is 4. | |

UL-equivalent group | upper central series and lower central series coincide. | |||

stem group | the center is contained in the derived subgroup | Yes | follows from being a non-abelian UL-equivalent group. | |

directly indecomposable group | nontrivial and cannot be expressed as an internal direct product of nontrivial subgroups | Yes | ||

centrally indecomposable group | nontrivial and cannot be expressed as a internal central product of proper nontrivial subgroups | Yes | ||

splitting-simple group | nontrivial and cannot be expressed as an internal semidirect product of nontrivial subgroups | Yes |

## Subgroups

`Further information: Subgroup structure of generalized quaternion group:Q16`

- The trivial subgroup. Isomorphic to trivial group. (1)
- The center, which is a subgroup of order two, generated by . Isomorphic to cyclic group:Z2. (1)
- The cyclic subgroup of order four generated by . Isomorphic to cyclic group:Z4. (1)
- The four cyclic subgroups of order four, namely: , , and . These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising and and the other comprising and . Isomorphic to cyclic group:Z4. (4)
- The cyclic subgroup of order eight, generated by . This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
- Two quaternion groups of order eight, namely and . Isomorphic to quaternion group. (2)
- The whole group. (1)

## Subgroup-defining functions

Subgroup-defining function | Subgroup type in list | Page on subgroup embedding | Isomorphism class | Comment |
---|---|---|---|---|

Center | (2) | cyclic group:Z2 | ||

Commutator subgroup | (3) | cyclic group:Z4 | ||

Frattini subgroup | (3) | cyclic group:Z4 | ||

Socle | (2) | cyclic group:Z2 | ||

Join of abelian subgroups of maximum order | (5) | cyclic group:Z8 | ||

Join of abelian subgroups of maximum rank | (7) | whole group | ||

Join of elementary abelian subgroups of maximum order | (2) | cyclic group:Z2 |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 9 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(16,9)`

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

`gap> G := SmallGroup(16,9);`

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

`IdGroup(G) = [16,9]`

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 14 among the groups of order 16. This information can be used to construct the group in GAP using the Gap3CatalogueGroup function as follows:

`Gap3CatalogueGroup(16,14)`

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(16,14);`

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

`Gap3CatalogueIdGroup(G) = [16,14]`

or just do:

`Gap3CatalogueIdGroup(G)`

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

### Description by presentation

gap> F := FreeGroup(2); <free group on the generators [ f1, f2 ]> gap> G := F/[F.1^8,F.2^2*F.1^(-4),F.2*F.1*F.2^(-1)*F.1]; <fp group on the generators [ f1, f2 ]> gap> IdGroup(G); [ 16, 9 ]

### Other descriptions

Description | Functions used |
---|---|

SylowSubgroup(SL(2,7),2) |
SylowSubgroup, SL |