Quaternion group

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Definition

Definition by presentation

The quaternion group has the following presentation:

$\langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$

The identity is denoted $1$, the common element $i^2 = j^2 = k^2 = ijk$ is denoted $-1$, and the elements $i^3, j^3, k^3$ are denoted $-i,-j,-k$ 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 $\Z/4\Z$.
• It is the holomorph of the cyclic group of order 4.
• It is the group comprising eight elements $1,-1,i,-i,j,-j,k,-k$ where 1 is the identity element, $(-1)^2 = 1$ and all the other elements are squareroots of $-1$, such that $(-1)i = -i, (-1)j = -j, (-1)k= -k$ and further, $ij = k, ji = -k, jk = i, kj = -1, ki = j ik = -j$ (the remaining relations can be deduced from these).
• It is the dicyclic group with parameter 2, viz $Dic_2$.
• It is the Fibonacci group $F(2,3)$.

Multiplication table

Element $\! 1$ $\! -1$ $\! i$ $\! -i$ $\! j$ $\! -j$ $\! k$ $\! -k$
$\! 1$ $\! 1$ $\! -1$ $\! i$ $\! -i$ $\! j$ $\! -j$ $\! k$ $\! -k$
$\! -1$ $\! -1$ $\! 1$ $\! -i$ $\! i$ $\! -j$ $\! j$ $\! -k$ $\! k$
$\! i$ $\! i$ $\! -i$ $\! -1$ $\! 1$ $\! k$ $\! -k$ $\! -j$ $\! j$
$\! -i$ $\! -i$ $\! i$ $\! 1$ $\! -1$ $\! -k$ $\! k$ $\! j$ $\! -j$
$\! j$ $\! j$ $\! -j$ $\! -k$ $\! k$ $\! -1$ $\! 1$ $\! i$ $\! -i$
$\! -j$ $\! -j$ $\! j$ $\! k$ $\! -k$ $\! 1$ $\! -1$ $\! -i$ $\! i$
$\! k$ $\! k$ $\! -k$ $\! j$ $\! -j$ $\! -i$ $\! i$ $\! -1$ $\! 1$
$\! -k$ $\! -k$ $\! k$ $\! -j$ $\! j$ $\! i$ $\! -i$ $\! 1$ $\! -1$

Families

1. The construction of the quaternion group can be mimicked for other primes giving, in general, a non-Abelian group of order $p^3$. The general construction involves taking a semidirect product of the cyclic group of order $p^2$ with a subgroup of order $p$ in the automorphism group, say the subgroup generated by the automorphism taking an element to its $(p+1)^{th}$.
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 $p$-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 number 5 among groups of order 8
Hall-Senior symbol $8\Gamma_2a_2$

Elements

Further information: Element structure of quaternion group

Conjugacy class structure

Conjugacy class Size of conjugacy class Order of elements in conjugacy class Centralizer of first element of class
$\! \{ 1 \}$ 1 1 whole group
$\! \{ -1 \}$ 1 2 whole group
$\! \{ i,-i \}$ 2 4 $\{ 1,-1,i,-i \}$, same as $\langle i \rangle$
$\! \{ j,-j \}$ 2 4 $\{ 1,-1,j,-j\}$ -- same as $\langle j \rangle$
$\! \{ k,-k \}$ 2 4 $\{ 1,-1,k,-k \}$ -- same as $\langle k \rangle$

Automorphism class structure

Equivalence class (orbit) under action of automorphisms Size of equivalence class (orbit) Number of conjugacy classes in it Size of each conjugacy class Order of elements
$\! \{ 1 \}$ 1 1 1 1
$\! \{ -1 \}$ 1 1 1 2
$\! \{ i,-i,j,-j,k,-k \}$ 6 3 2 4

Arithmetic functions

Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions
Function Value Similar groups Explanation
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 8 groups with same order
prime-base logarithm of order 3 groups with same prime-base logarithm of order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group Cyclic subgroup of order four.
prime-base logarithm of exponent 2 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 2 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
Frattini length 2 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 Generators of two cyclic subgroups of order four.
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 All proper subgroups are cyclic.
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.

Arithmetic functions of an element-counting nature

Further information: element structure of quaternion group

Function Value Similar groups Explanation
number of conjugacy classes 5 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes See element structure of dicyclic groups.
number of equivalence classes under real conjugacy 5 groups with same order and number of equivalence classes under real conjugacy | groups with same number of equivalence classes under real conjugacy Same as number of conjugacy classes, because the group is an ambivalent group.
number of conjugacy classes of real elements 5 groups with same order and number of conjugacy classes of real elements | groups with same number of conjugacy classes of real elements Same as number of conjugacy clases, because the group is an ambivalent group.
number of equivalence classes under rational conjugacy 5 groups with same order and number of equivalence classes under rational conjugacy | groups with same number of equivalence classes under rational conjugacy Same as number of conjugacy classes, because the group is a rational group (though not a rational representation group).
number of conjugacy classes of rational elements 5 groups with same order and number of conjugacy classes of rational elements | groups with same number of conjugacy classes of rational elements Same as number of conjugacy classes, because the group is a rational group (though not a rational representation group).

Arithmetic functions of a subgroup-counting nature

Further information: subgroup structure of quaternion group

Function Value Similar groups Explanation
number of subgroups 6
number of conjugacy classes of subgroups 6
number of normal subgroups 6 groups with same order and number of normal subgroups | groups with same number of normal subgroups
number of automorphism classes of subgroups 4

Lists of numerical invariants

List Value Explanation/comment
conjugacy class sizes $1,1,2,2,2$ $\pm i, \pm j, \pm k$ are each conjugacy classes of non-central elements.
degrees of irreducible representations $1,1,1,1,2$ See linear representation theory of quaternion group
order statistics $1 \mapsto 1, 2 \mapsto 1, 4 \mapsto 6$
orders of subgroups $1,2,4,4,4,8$ See subgroup structure of quaternion group

Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 8#Group 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 $i$ and $j$ don't commute Smallest non-abelian group of prime power order
metacyclic group Yes Cyclic normal subgroup of order four, cyclic quotient of order two
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

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) Nilpotency class
trivial subgroup $\{ 1 \}$ trivial subgroup 1 8 1 1 1 quaternion group 0
center of quaternion group $\{ 1, -1\}$ cyclic group:Z2 2 4 1 1 1 Klein four-group 1
cyclic maximal subgroups of quaternion group $\{ 1,-1,i,-i \}$
$\{ 1,-1,j,-j \}$
$\{ 1,-1,k,-k \}$
cyclic group:Z4 4 2 3 1 3 cyclic group:Z2 1
whole group $\{ 1,-1,i,-i,j,-j,k,-k \}$ quaternion group 8 1 1 1 1 trivial group 2
Total (4 rows) -- -- -- -- 6 -- 6 -- --

Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function What it means Value as subgroup Value as group Order Associated quotient-defining function Value as group Order (= index of subgroup)
center elements that commute with every group element center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2 inner automorphism group Klein four-group 4
derived subgroup subgroup generated by all commutators center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2 abelianization Klein four-group 4
Frattini subgroup intersection of all maximal subgroups center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2 Frattini quotient Klein four-group 4
Jacobson radical intersection of all maximal normal subgroups center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2  ? Klein four-group 4
socle join of all minimal normal subgroups center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2  ? Klein four-group 4
Fitting subgroup join of all nilpotent normal subgroups whole group quaternion group 8 Fitting quotient trivial group 1
join of abelian subgroups of maximum order join of all abelian subgroups of maximum order among abelian subgroups whole group quaternion group 8  ? trivial group 1
join of abelian subgroups of maximum rank join of all abelian subgroups of maximum rank among abelian subgroups whole group quaternion group 8  ? trivial group 1
join of elementary abelian subgroups of maximum order join of all elementary abelian subgroups of maximum order among elementary abelian subgroups center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2  ? Klein four-group 4
ZJ-subgroup center of the join of abelian subgroups of maximum order center of quaternion group: $\{ 1, -1 \}$ cyclic group:Z2 2  ? Klein four-group 4

Automorphisms and endomorphisms

Further information: endomorphism structure of quaternion group

Construct Value Order Second part of GAP ID (if group)
endomorphism monoid  ?  ? --
automorphism group symmetric group:S4 24 12
inner automorphism group Klein four-group 4 2
outer automorphism group symmetric group:S3 6 1
group of class-preserving automorphisms Klein four-group 4 2
group of IA-automorphisms Klein four-group 4 2
quotient of class-preserving automorphism group by inner automorphism group trivial group 1 1
quotient of IA-automorphism group by inner automorphism group trivial group 1 1
group of center-fixing automorphisms symmetric group:S4 24 12
extended automorphism group direct product of S4 and Z2 48 48
holomorph  ? 192
inner holomorph inner holomorph of D8 ($D_8$ and the quaternion group have the same holomorph) 32 49

Distinguishing features

Smallest of its kind

• This is a non-abelian nilpotent group of smallest possible order, along with dihedral group:D8.
• This is a non-abelian Dedekind group (or Hamiltonian group) of smallest possible order. Dedekind means that every subgroup is normal.

Different from others of the same order

• It is the only non-abelian Dedekind group of its order.
• It is the only non-abelian T-group of its order.
• It is the only group of its order for which the rank (in the sense of the maximum possible rank of an abelian subgroup) is strictly smaller than the minimum size of generating set: For this group, the former is 1 and the latter is 2.

GAP implementation

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 $G$:

gap> G := SmallGroup(8,4);

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

Short descriptions

Description Functions used Mathematical comment
SylowSubgroup(SL(2,3),2) SylowSubgroup and SL The $2$-Sylow subgroup of special linear group:SL(2,3)
ExtraspecialGroup(2^3,'-') ExtraspecialGroup The extraspecial group of order $2^3$ and '-' type
SylowSubgroup(SL(2,5),2) SylowSubgroup and SL The $2$-Sylow subgroup of special linear group:SL(2,5)