M
Group Theory
M.1 Definition and simple examples
A group G is a set of elements (a, b, c, . . .) with a law for combining any two
elements a, b so as to form their ordered ‘product’ ab, such that the following
four conditions hold:
(i) For every a, b ∈ G, the product ab ∈ G (the symbol ‘∈’ means ‘belongs
to’, or ‘is a member of’).
(ii) The law of combination is associative, i.e.
(ab)c = a(bc).
(M.1)
(iii) G contains a unique identity element, e, such that for all a ∈ G,
ae = ea = a.
(M.2)
(iv) For all a ∈ G, there is a unique inverse element, a
−1 , such that
aa
−1 = a
−1 a = e.
(M.3)
Note that in general the law of combination is not commutative – i.e.
ab = ba; if it is commutative, the group is Abelian; if not, it is non-Abelian.
Any finite set of elements satisfying the conditions (i)–(iv) forms a finite group,
the order of the group being equal to the number of elements in the set. If
the set does not have a finite number of elements it is an infinite group.
As a simple example, the set of four numbers (1, i, –1, –i) form a finite
Abelian group of order 4, with the law of combination being ordinary multiplication. The reader may check that each of (i)–(iv) is satisfied, with e taken
to be the number 1, and the inverse being the algebraic reciprocal. A second
group of order 4 is provided by the matrices
(
) (
) (
) (
)
1 0
0 1
,
,
−1 0
0
,
−1 ,
(M.4)
0 1
−1 0
0 −1
1 0
with the combination law being matrix multiplication, ‘e’ being the first (unit)
matrix, and the inverse being the usual matrix inverse. Although matrix
435
/
Group Theory
M.1 Definition and simple examples
A group G is a set of elements (a, b, c, . . .) with a law for combining any two
elements a, b so as to form their ordered ‘product’ ab, such that the following
four conditions hold:
(i) For every a, b ∈ G, the product ab ∈ G (the symbol ‘∈’ means ‘belongs
to’, or ‘is a member of’).
(ii) The law of combination is associative, i.e.
(ab)c = a(bc).
(M.1)
(iii) G contains a unique identity element, e, such that for all a ∈ G,
ae = ea = a.
(M.2)
(iv) For all a ∈ G, there is a unique inverse element, a
−1 , such that
aa
−1 = a
−1 a = e.
(M.3)
Note that in general the law of combination is not commutative – i.e.
ab = ba; if it is commutative, the group is Abelian; if not, it is non-Abelian.
Any finite set of elements satisfying the conditions (i)–(iv) forms a finite group,
the order of the group being equal to the number of elements in the set. If
the set does not have a finite number of elements it is an infinite group.
As a simple example, the set of four numbers (1, i, –1, –i) form a finite
Abelian group of order 4, with the law of combination being ordinary multiplication. The reader may check that each of (i)–(iv) is satisfied, with e taken
to be the number 1, and the inverse being the algebraic reciprocal. A second
group of order 4 is provided by the matrices
(
) (
) (
) (
)
1 0
0 1
,
,
−1 0
0
,
−1 ,
(M.4)
0 1
−1 0
0 −1
1 0
with the combination law being matrix multiplication, ‘e’ being the first (unit)
matrix, and the inverse being the usual matrix inverse. Although matrix
435
/