GROUP THEORY
1 i − 1 −i
1 1 i − 1 −i
i i − 1 −i 1
− 1 − 1 −i 1 i
−i −i 1 i − 11243
1 1243
2 2431
4 4312
3 3124
Table 28.5 A comparison between tables 28.4 and 28.2(b), the latter with its
columns reordered.IABC
I IABC
A ABC I
B BC I A
C CIAB
Table 28.6 The common structure exemplified by tables 28.4 and 28.2(b), the
latter with its columns reordered.amount of relabelling (or, equivalently, no allocation of the symbolsA, B, C,amongst
i,− 1 ,−i) can bring table 28.4 into the form of table 28.3. We conclude that the group
{ 1 ,i,− 1 ,−i}is not isomorphic toSorS′. An alternative way of stating the observation is
to say that the group contains only one element of order 2 whilst a group corresponding
to table 28.3 contains three such elements.
However, if the rows and columns of table 28.2(b) – in which the identity does appear
twice on the diagonal and which therefore has the potential to be equivalent to table 28.4 –
are rearranged by making the heading order 1, 2 , 4 ,3 then the two tables can be compared
in the forms shown in table 28.5. They can thus be seen to have the same structure, namely
that shown in table 28.6.
We therefore conclude that the group of four elements{ 1 ,i,− 1 ,−i}under ordinary mul-
tiplication of complex numbers is isomorphic to the group{ 1 , 2 , 3 , 4 }under multiplication
(mod 5).
What we have done does not prove it, but the two tables 28.3 and 28.6 are infact the only possible tables for a group of order 4, i.e. a group containing exactly
four elements.
28.3 Non-Abelian groupsSo far, all the groups for which we have constructed multiplication tables have
been based on some form of arithmetic multiplication, a commutative operation,
with the result that the groups have been Abelian and the tables symmetric
about the leading diagonal. We now turn to examples of groups in which some
non-commutation occurs. It should be noted, in passing, that non-commutation
cannotoccurthroughouta group, as the identity always commutes with any
element in its group.