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 | ![]() |
|
metacyclic group | Yes | ![]() |
|
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 |