Men of Mathematics (54 page)

In proving a proposition by the indirect method, a contradiction is deduced from the assumed falsity of the proposition; whence it follows, in Aristotelian logic, that the proposition is true. Cauchy could not meet the objection by supplying direct proofs, and Malus gave in—still unconvinced that Cauchy had proved anything. When we come to the conclusion of the whole story (in the last chapter) we shall see the same objection being raised in other connections by the intuitionists. If Malus failed to make Cauchy see the point in 1812, Malus was avenged by Brouwer in 1912 and thereafter when Brouwer succeeded in making some of Cauchy's successors in mathematical analysis at least see that there is a point to be seen. Aristotelian logic, as Malus was trying to tell Cauchy, is not always a safe method of reasoning in mathematics.

Passing to the
theory of substitutions,
begun systematically by Cauchy, and elaborated by him in a long series of papers in the middle 1840's, which developed into the
theory of finite groups,
we shall presently illustrate the underlying notions by a simple example. First, however, the leading properties of a
group of operations
may be described informally.

Operations will be denoted by capital letters,
A, B, C, D, . . .;
and the performance of two operations
in succession,
say
A first, B second,
will be indicated by juxtaposition thus,
AB.
Note that
BA,
by what has just been said, means that
B
is performed first,
A
second; so that
AB
and
BA
are
not necessarily
the same operation. For example, if
A
is the operation “add 10 to a given number,” and
B
is the operation “divide a given number by 10,”
AB
applied to
x
gives
while
BA
gives
or
and the resulting fractions are unequal; hence
AB
and
BA
are distinct.

If the effects of two operations
X, T
are the same,
X
and
T
are said to be
equal
(or
equivalent),
and this is expressed by writing
X
=
Y.

The next fundamental notion is that of
associativity.
If for
every
triple of operations, say
U, V, W
is any triple, in the set,
(UV)W = U(VW),
the set is said to satisfy the
associative
law. By (
UV)W
is meant that
UV
is performed first, then, on the result,
W
is performed; by
U(VW)
is meant that
U
is performed first, then, on the result of this
VW
is performed.

The last fundamental notion is that of an
identical operation,
or an
identity:
an operation I which leaves unchanged whatever it operates on is called an
identity.

With these notions we can now state the simple postulates which define a group of operations.

A set of operations
I
,
A, B, C, . . . , X, T, . . .
is said to form a
group
if the postulates (1) –
(4)
are satisfied.

(1) There is a rule of combination applicable to
any
pair
X, Y
of operations
^I
in the set such that the result, denoted by
XY,
of combining
X, Y,
in this order, according to the rule of combination, is a uniquely determined operation in the set.

(2) For
any three
operations
X, Y, Z
in the set, the rule in (1) is associative, namely
(XY)Z = X(YZ).

(3) There is a unique identity
I
in the set, such that, for every operation
X
in the set,
IX = XI = X.

(4) If
X
is
any
operation in the set, there is in the set a
unique
operation, say
X′,
such that
XX′
=
I
(it can be easily proved that
X′X
=
i
also).

These postulates contain redundancies deducible from other statements in (1) - (4), but in the form given the postulates are easier to grasp. To illustrate a group we shall take a very simple example relating to
permutations
(arrangements) of letters. This may seem trivial, but such
permutation
or
substitution
groups were found to be the long-sought clue to the algebraic solvability of equations.

There are precisely
6
orders in which the 3 letters
a, b, c
can be written, namely
abc, acb, bca, bac, cab, cba.
Take any one of these, say the first
abc,
as the initial order. By what permutations of the letters can we pass from this to the remaining 5 arrangements? To pass from
abc
to
acb
it is sufficient to
interchange,
or
permute, b
and
c.
To indicate the
operation
of permuting
b
and
c,
we write
(be),
which is read,
“b
into
c,
and
c
into
b.”
From
abc
to
bca
we pass by
a
into
b, b
into
c,
and
c
into
a,
which is written
(abc).
The order
abc
itself is obtained from
abc
by
no
change, namely
a
into
a, b
into
b, c
into
c,
which
is the
identity
substitution and is denoted by
I
. Proceeding similarly with all
6
orders

abc, acb, bca, bac, cab, cba,

we get the corresponding
substitutions,

I, (bc), (abc), (ab), (acb), (ac).

The “rule of combination” in the postulates is here as follows. Take any two of the substitutions, say
(be)
and
(acb),
and consider the effect of these applied successively in the order stated, namely
(be)
first and
(acb)
second:
(be)
carries
b
into
c,
then
(acb)
carries
c
into
b.
Thus
b
is left as it was. Take the next letter,
c,
in
(be):
by
(be), c
is carried into
b,
which, by
(acb)
is carried into
a;
thus
c
is carried into
a.
Continuing, we see what
a
is now carried into:
(be)
leaves
a
as it was, but
(acb)
carries
a
into
c.
Finally then the total effect of
(be)
followed by
(acb)
is seen to be
(ca),
which we indicate by writing
(be)(acb) = (ca) = (ac).

In the same way it is easily verified that

(acb) (abc) = (abc) (acb)
=
I
;

(abc)(ac) = (ab); (bc)(ac) = (acb),

and so on for all possible pairs. Thus postulate (1) is satisfied for these
6
substitutions, and it can be checked that (2), (3), (4) are also satisfied.

All this is summed up in the “multiplication table” of the group, which we shall write out, denoting the substitutions by the letters under them (to save space),

I, (be), (abc), (ab), (acb), (ac)

I, A, B, C, D, E.

In reading the table any letter, say C, is taken from the left-hand
column,
and any letter, say
D,
from the top
row,
and the entry, here
A,
where the corresponding
row
and
column
intersect is the result of
CD.
Thus
CD = A, DC = E, EA = B,
and so on.

As an example we may verify the associative law for
(AB)C
and
A(BC),
which should be equal. First,
AB
= C; hence
(AB)C = CC
=
I
. Again
BC = A;
hence
A(BC) = AA
=
I
. In the same way
A(DB) = AI = A; (AD)B = EB = A;
thus
(AD)B = A(DB).

The total number of distinct operations in a group is called its
order.
Here
6
is the order of the group. By inspection of the table we pick out several
subgroups,
for example, which are of the respective orders 1, 2,
3.
This illustrates one of the fundamental theorems proved by Cauchy:
the order of any subgroup is a divisor of the order of the group.