Difference between revisions of "Quaternion group"

Revision as of 19:47, 26 March 2010

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

Definition by presentation

The quaternion group has the following presentation:

The identity is denoted , the common element is denoted , and the elements are denoted respectively.

Verbal definitions

The quaternion group is a group with eight elements, which can be described in any of the following ways:

• It is the holomorph of the ring .
• It is the holomorph of the cyclic group of order 4.
• It is the group comprising eight elements where 1 is the identity element, and all the other elements are squareroots of , such that and further, (the remaining relations can be deduced from these).
• It is the dicyclic group with parameter 2, viz .
• It is the Fibonacci group .

Multiplication table

Element

Families

1. The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order . The general construction involves taking a semidirect product of the cyclic group of order with a subgroup of order in the automorphism group, say the subgroup generated by the automorphism taking an element to its .
2. The quaternion group also generalizes to the family of dicyclic groups (also known as binary dihedral groups) and also to the family of generalized quaternion groups (which are the dicyclic groups whose order is a power of 2).
3. The quaternion group is part of a larger family of -groups called extraspecial groups. An extraspecial group is a group of prime power order whose center, commutator subgroup and Frattini subgroup coincide, and are all cyclic of prime order.

Position in classifications

Type of classification	Name in that classification
GAP ID	(8,4), i.e., the 4th among the groups of order 8
Hall-Senior classification

Elements

Upto conjugacy

The quaternion group has five conjugacy classes:

1. The identity element: This has order 1 and size 1
2. The element : This has order 2 and size 1
3. The two-element conjugacy class comprising : This has order 4 and size 2
4. The two-element conjugacy class comprising : This has order 4 and size 2
5. The two-element conjugacy class comprising : This has order 4 and size 2

Upto automorphism

Under the action of automorphisms, the last three conjugacy classes get merged, so there are three equivalence classes, of sizes 1, 1, and 6.

Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 8#Arithmetic functions.

Lists of numerical invariants

List	Value	Explanation/comment
conjugacy class sizes		are each conjugacy classes of non-central elements.
degrees of irreducible representations		See linear representation theory of quaternion group
order statistics
orders of subgroups		See subgroup structure of quaternion group

Group properties

Want to compare with other groups of the same order? Check out groups of order 8#Group properties.

Property	Satisfied	Explanation	Comment
abelian group	No	and don't commute	Smallest non-abelian group of prime power order
nilpotent group	Yes	Prime power order implies nilpotent	Smallest nilpotent non-abelian group, along with dihedral group:D8.
metacyclic group	Yes	Cyclic normal subgroup of order four, cyclic quotient of order two
supersolvable group	Yes	Metacyclic implies supersolvable
solvable group	Yes	Metacyclic implies solvable
Dedekind group	Yes	Every subgroup is normal	Smallest non-abelian Dedekind group
T-group	Yes	Dedekind implies T-group
monolithic group	Yes	Unique minimal normal subgroup of order two
one-headed group	No	Three distinct maximal normal subgroups of order four
SC-group	No
ACIC-group	Yes	Every automorph-conjugate subgroup is characteristic
ambivalent group	Yes
rational group	Yes	Any two elements that generate the same cyclic group are conjugate	Thus, all characters are integer-valued.
rational-representation group	Yes	A two-dimensional representation that is not rational.	Contrast with dihedral group:D8, that is rational-representation.
maximal class group	Yes
group of nilpotency class two	Yes
extraspecial group	Yes
special group	Yes
Frattini-in-center group	Yes
Frobenius group	No	Frobenius groups are centerless, and this group isn't.
Camina group	Yes	extraspecial implies Camina
group in which every element is automorphic to its inverse	Yes	Follows from being an ambivalent group
group in which any two elements generating the same cyclic subgroup are automorphic	Yes	Follows from being a rational group
group in which every element is order-automorphic	Yes
directly indecomposable group	Yes
centrally indecomposable group	Yes
splitting-simple group	Yes

Subgroups

Further information: Subgroup structure of quaternion group

Lattice of subgroups of the quaternion group

The quaternion group has six subgroups:

1. The trivial subgroup (1)
2. The center, which is the unique minimal subgroup. This is a two-element subgroup comprising (1)
3. The three cyclic subgroups of order four, generated by respectively. These are all normal, but are automorphs of each other (3)
4. The whole group (1)

Normal subgroups

All subgroups are normal. The subgroups are the whole group, the trivial subgroup, the center, and three copies of the cyclic group on 4 elements. This makes the quaternion group a Dedekind group.

Characteristic subgroups

There are only three characteristic subgroups: the whole group, the trivial subgroup and the center.

Subgroup-defining functions

Quotient-defining functions

Other associated constructs

Associated construct	Value (isomorphism class)	Comment
Automorphism group	symmetric group:S4
Outer automorphism group	symmetric group:S3
Inner holomorph	inner holomorph of D8	The inner holomorphs of and the quaternion group are isomorphic.

Supergroups

Further information: Supergroups of quaternion group

Implementation in GAP

Group ID

This finite group has order 8 and has ID 4 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(8,4)

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

gap> G := SmallGroup(8,4);

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

IdGroup(G) = [8,4]

or just do:

IdGroup(G)

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

Other descriptions

Internal links

• Linear representation theory of quaternion group
• Subgroup structure of quaternion group
• Supergroups of quaternion group
• Fusion systems for quaternion group