The Quaternion Group Q

The Quaternion Group Q

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 27 January 2011

The quaternion group Q

The quaternion group Q is as in the following table. The element -1 acts like -1 in the complex numbers, it takes everything to its negative, and the negative of a negative is a positive.

Set Operation
Q= 1,-1,i,-i,j,-j, k,-k i2=j2=k2= ijk=-1

The complete multiplication table for Q is as follows.

Multiplication table
Q 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

Center Abelian Conjugacy classes Subgroups
ZQ= 1,-1 No 𝒞1 = 1 H0= Q
𝒞-1 = -1 H1= ±1,±i
𝒞i = ±i H2= ±1,±j
𝒞j = ±j H3= ±1,±k
𝒞k = ±k H4= ±1
H5= 1

Element g Order οg Centralizer Zg Conjugacy Class 𝒞g
1 1 Q 𝒞1
-1 2 Q 𝒞-1
i 4 H1 𝒞i
-i 4 H1 𝒞i
j 4 H2 𝒞j
-j 4 H2 𝒞j
k 4 H3 𝒞k
-k 4 H3 𝒞k

Generators Relations Realization
S,T S2=T2= ST2 S=i,T=j,ST=k

Subgroups Hi Structure Index Normal Quotient group
H0= Q H0=Q Q:Q = 1 Yes Q /H0 1
H1= ±1,±i H1 μ4 Q:H1 = 2 Yes Q /H1 μ2
H2= ±1,±j H2 μ4 Q:H2 =2 Yes Q /H2 μ2
H3= ±1,±k H3 μ4 Q:H3 =2 Yes Q /H3 μ2
H4= ±1 H4μ2 Q:H4 =4 Yes Q /H4 μ2×μ2
H9= 1 H9=1 Q:1 =8 Yes Q /H9 Q

Orders Inclusions 8 4 2 1 Q ±1,±i ±1,±j ±1,±k ±1 1

Subgroups Hi Left Cosets Right Cosets
H0=Q Q Q
H1 = ±1,±i H1= ±1,±i H1= ±1,±i
jH1= ±j,±k H1j= ±j,±k
H2 = ±1,±j H2= ±1,±j H2= ±1,±j
iH2= ±i,±k H2j= ±i,±k
H3 = ±1,±k H3= ±1,±k H3= ±1,±k
iH3= ±i,±j H3i= ±i,±j
H4= ±1 H4= ±1 H4= ±1
iH4= ±i H4i= ±i
jH4= ±j H4j= ±j
kH4= ±k H4k= ±k
H5= 1 H5= 1 H5= 1
-1H5= -1 H5-1= -1
iH5= i H5i= i
-iH5= -i H5-i= -i
jH5= j H5j= j
-jH5= -j H5-j= -j
kH5= k H5k= k
-kH5= k H5-k= -k

Subgroups Hi Normalizer NHi Centralizer ZHi
H0=Q Q ZQ= H4= ±1
H1= i Q H1= i
H2= j Q H2= j
H3= k Q H3= k
H4= ±1 Q Q
H5= 1 Q Q

Homomorphism Kernel Image
ϕ0: Q 1 i 1 j 1 kerϕ0= Q imϕ0=1
ϕ1: Q μ2 i 1 j -1 kerϕ1= H1= ±1,±i imϕ1 =μ2
ϕ2: Q μ2 i -1 j 1 kerϕ2 =H2 = ±1,±j imϕ2 =μ2
ϕ3: Q μ2 i -1 j -1 kerϕ3 =H2 = ±1,±k imϕ3 =μ2
ϕ4: Q GL2 i i 0 0 i j 0 1 -1 0 k 0 i i 0 kerϕ4 =H5= 1 imϕ4= ±1 0 0 ±1 , ±i 0 0 i , 0 ±1 1 0 , 0 ±i i 0
ϕ5: Q μ2×μ2 i -1,1 j 1,-1 kerϕ5 =H4 = ±1 imϕ5 =μ2×μ2

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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