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 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 = -i, 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

In the table below, the row element is multiplied on the left and the column element on the right.

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


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.
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