Generalized quaternion group:Q16

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Definition

The group $Q_{16}$, sometimes termed the generalized quaternion group of order $16$, is a generalized quaternion group. It can be described by the following presentation:

$G:=⟨a,b,c\mid a^4=b^2=c^2=abc⟩$.

Note that $c=ab=b{a}^{-1}$ from these relations, and $ba{b}^{-1}=a^{-1}$. This in turn forces that $b^2=b(b^2)b^{-1}=b{a}^4{b}^{-1}=a^{-4}=b^{-2}$, forcing $b^2=a^4=c^2=abc$ to have order two. We shall denote this element of order two, which is clearly central, as $z$.

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

$G:=⟨a,b\mid a^4=b^2=abab⟩$

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

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group | groups with same exponent of a group	cyclic subgroup of order 8.
prime-base logarithm of exponent	3	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	3	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length	the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length	3	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group	1	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	all abelian subgroups are cyclic.
normal rank of a p-group	1	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	all abelian normal subgroups are cyclic.
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	All abelian characteristic subgroups are cyclic.

Subgroups

Further information: Subgroup structure of generalized quaternion group:Q16

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The center, which is a subgroup of order two, generated by $z=a^4=b^2=c^2$. Isomorphic to cyclic group:Z2. (1)
3. The cyclic subgroup of order four generated by $a^2$. Isomorphic to cyclic group:Z4. (1)
4. The four cyclic subgroups of order four, namely: $⟨b⟩$, $⟨ab⟩$, $⟨a^{2}b⟩$ and $⟨a^{3}b⟩$. These come in two conjugacy classes of 2-subnormal subgroups, one conjugacy class comprising $⟨ab⟩$ and $⟨a^{3}b⟩$ and the other comprising $⟨b⟩$ and $⟨a^{2}b⟩$. Isomorphic to cyclic group:Z4. (4)
5. The cyclic subgroup of order eight, generated by $a$. This is characteristic; in fact, it equals the centralizer of commutator subgroup. Isomorphic to cyclic group:Z8. (1)
6. Two quaternion groups of order eight, namely $⟨a^2,b⟩$ and $⟨a^2,ab⟩$. Isomorphic to quaternion group. (2)
7. 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 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 $G$:

gap> G := SmallGroup(16,9);

Conversely, to check whether a given group $G$ 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.