500 14 Groups and group representations
14.1.2 Elementary properties
Here are the basic properties of groups that we need:
(i) Subgroups: If a subset of elements of a group forms a group, it is called a subgroup.
For example, Z
12
has a subgroup consisting of {0, 3, 6, 9}. Any group G possesses
at least two subgroups: the entirety of G itself, and the subgroup containing only the
identity element {e}. These are known as the trivial subgroups. Any other subgroups
are called proper subgroups.
(ii) Cosets: Given a subgroup H ⊆ G, having elements {h
1
, h
2
, ...}, and an element
g ∈ G, we form the (left) coset gH ={gh
1
, gh
2
, ...}. If two cosets g
1
H and g
2
H
intersect, they coincide. (Proof: if g
1
h
1
= g
2
h
2
, then g
2
= g
1
(h
1
h
−1
2
) and so
g
1
H = g
2
H .) If H is a finite group, each coset has the same number of distinct
elements as H . (Proof: if gh
1
= gh
2
then left multiplication by g
−1
shows that
h
1
= h
2
.) If the order of G is also finite, the group G is decomposed into an integer
number of cosets,
G = g
1
H + g
2
H +···, (14.1)
where “+” denotes the union of disjoint sets. From this we see that the order of H
must divide the order of G. This result is called Lagrange’s theorem. The set whose
elements are the cosets is denoted by G/H .
(iii) Normal subgroups: A subgroup H ={h
1
, h
2
, ...} of G is said to be normal,or
invariant,ifg
−1
Hg = H for all g ∈ G. This notation means that the set of
elements g
−1
Hg ={g
−1
h
1
g, g
−1
h
2
g, ...} coincides with H , or equivalently that
the map h (→ g
−1
hg does not take h ∈ H out of H , but simply scrambles the order
of the elements of H.
(iv) Quotient groups: Given a normal subgroup H , we can define a multiplication rule
on the set of cosets G/H ≡{g
1
H , g
2
H , ...} by taking a representative element
from each of g
i
H , and g
j
H , taking the product of these elements, and defining
(g
i
H )(g
j
H ) to be the coset in which this product lies. This coset is independent
of the representative elements chosen (this would not be so were the subgroup not
normal). The resulting group is called the quotient group of G by H , and is denoted
by G/H . (Note that the symbol “G/H ” is used to denote both the set of cosets, and,
when it exists, the group whose elements are these cosets.)
(v) Simple groups: A group G with no normal subgroups is said to be sim-
ple. The finite simple groups have been classified. They fall into various
infinite families (cyclic groups, alternating groups, 16 families of Lie type)
together with 26 sporadic groups, the largest of which, the Monster, has
order 808,017,424,794,512,875,886,459,904,961,710,757,005,754, 368,000,000,
000. The mysterious “Monstrous moonshine” links its representation theory to the
elliptic modular function J (τ ) and to string theory.
(vi) Conjugacy and conjugacy classes: Two group elements g
1
, g
2
are said to be con-
jugate in G if there is an element g ∈ G such that g
2
= g
−1
g
1
g.Ifg
1
is conjugate