GROUP THEORY
It is clear that cyclic groups are always Abelian and that each element, apart
from the identity, has orderg, the order of the group itself.
28.1.2 Further examples of groupsIn this section we consider some sets of objects, each set together with a law of
combination, and investigate whether they qualify as groups and, if not, why not.
We have already seen that the integers form a group under ordinary addition,but it is immediately apparent that (even if zero is excluded) they donotdo
so under ordinary multiplication. Unity must be the identity of the set, but the
requisite inverse of any integern,namely1/n, does not belong to the set of
integers for anynother than unity.
Other infinite sets of quantities that do form groups are the sets of all realnumbers, or of all complex numbers, under addition, and of the same two sets
excluding 0 under multiplication. All these groups are Abelian.
Although subtraction and division are normally considered the obvious coun-terparts of the operations of (ordinary) addition and multiplication, they are not
acceptable operations for use within groups since the associative law, (28.1), does
not hold. Explicitly,
X−(Y−Z)=(X−Y)−Z,X÷(Y÷Z)=(X÷Y)÷Z.From within the field of all non-zero complex numbers we can select just thosethat have unit modulus, i.e. are of the formeiθwhere 0≤θ< 2 π,toforma
group under multiplication, as can easily be verified:
eiθ^1 ×eiθ^2 =ei(θ^1 +θ^2 ) (closure),
ei^0 = 1 (identity),
ei(2π−θ)×eiθ=ei^2 π≡ei^0 = 1 (inverse).Closely related to the above group is the set of 2×2 rotation matrices that take
the form
M(θ)=(
cosθ −sinθ
sinθ cosθ)where, as before, 0≤θ< 2 π. These form a group when the law of combination
is that of matrix multiplication. The reader can easily verify that
M(θ)M(φ)=M(θ+φ) (closure),
M(0) =I 2 (identity),
M(2π−θ)=M−^1 (θ) (inverse).HereI 2 is the unit 2×2 matrix.