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2.1
Groups: simple properties and further examples
Elementary properties of groups
We start constructing the group theory. As in any axiomatic construction, in the proof of
our first theorem, we are only allowed to use the definition of the group; in the proof of the
second theorem, we can use the definition and the first theorem; in the proof of the third
theorem, we can use the definition and the first two theorems, and so on.
Our first theorem about groups will establish cancellation laws. In real arithmetic we
know that 2a = 2b implies a = b. To prove this, we need only multiply both sides of the
equation 2a = 2b by 1/2, which is the multiplicative inverse of 2. We use the same trick to
establish the cancellation law for any group.
Theorem 2.1. If G is a group with binary operation ∗, then the left and right cancellation
laws hold in G. That is, for any a, b, c ∈ G, a ∗ b = a ∗ c implies b = c, and b ∗ a = c ∗ a
implies b = c.
Proof. Suppose a ∗ b = a ∗ c. Then by the axiom G3, there exists an inverse a . Multiplying
a ∗ b = a ∗ c by a on the left, we get
a ∗ (a ∗ b) = a ∗ (a ∗ c).
By the associative law,
(a ∗ a) ∗ b = (a ∗ a) ∗ c.
By the definition of a in G3, one has a ∗ a = e, so
e ∗ b = e ∗ c.
By the definition of e in G2, this implies b = c.
Similarly, from b ∗ a = c ∗ a one can deduce that b = c upon multiplication on the right
by a and use of the axioms of the group.
Our next theorem will enable us to solve ‘linear equations’ in a group. In real arithmetic,
we can solve the equation ax = b (with respect to the variable x) by multiplying both sides
by 1/a. The same idea works for groups. Note that now we can use use our first theorem.
Theorem 2.2. If G is a group with binary operation ∗, and a and b are any elements of
G, then the equations a ∗ x = b and y ∗ a = b have unique solutions on G.
Proof. First we prove that there exists at least one solution of a ∗ x = b. Indeed, let us
show that x = a ∗ b is a solution:
a ∗ (a ∗ b) = (a ∗ a ) ∗ b (associative law)
= e ∗ b (definition of a )
= b (property of e).
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Thus x = a ∗ b is a solution of a ∗ x = b. Similarly one shows that y = b ∗ a is a solution
of y ∗ a = b.
Now let us show that the solution x to x = a ∗ b is unique. As usually in the proof of
uniqueness, we have to assume that there are two solutions and prove that they coincide.
Suppose x1 and x2 are two solutions so that a ∗ x1 = b and a ∗ x2 = b. Then a ∗ x1 = a ∗ x2 ,
so, by Theorem 2.1, x1 = x2 . The uniqueness of y follows similarly.
Note that the solutions x = a ∗ b and y = b ∗ a need not be the same unless ∗ is
commutative.
Theorem 2.3. Let G be a group with binary operation ∗.
(i) There is only one identity e such that
e∗x=x∗e=x
for all x ∈ G.
(ii) For each a ∈ G, there is only one element a such that
a ∗ a = a ∗ a = e.
In summary, the identity and inverses are unique in a group.
Proof. (i) Suppose there are two elements e and e1 in the group such that
e ∗ x = x ∗ e = x,
e1 ∗ x = x ∗ e1 = x
(2.1)
(2.2)
for all x ∈ G; let us show that e = e1 . First let us put x = e1 in (2.1); we get
e ∗ e1 = e1 .
Next, put x = e in (2.2); we get
e ∗ e1 = e.
Clearly, the last two relations imply e = e1 .
(ii)Suppose that a ∈ G has inverses a and a so that a ∗ a = a ∗ a = e and a ∗ a =
a ∗ a = e. Then
a ∗ a = a ∗ a = e,
and using the left cancellation law (Theorem 2.1), we get a = a , so the inverse of a is
unique.
Theorem 2.4. If G is a group with binary operation ∗, and a and b are any elements of
G, then the inverse of a ∗ b is given by (a ∗ b) = b ∗ a .
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Proof. By Theorem 2.3, the inverse of a ∗ b is unique, so it is sufficient to prove that b ∗ a
satisfies
(a ∗ b) ∗ (b ∗ a ) = (b ∗ a ) ∗ (a ∗ b) = e.
But using associativity, the definition of an inverse and the property of e, we have
(a ∗ b) ∗ (b ∗ a ) = a ∗ (b ∗ b ) ∗ a = a ∗ e ∗ a = a ∗ a = e,
and in the same way (b ∗ a ) ∗ (a ∗ b) = e.
Notation and terminology From now on we will change a bit our notation and
terminology and follow the conventions used in group theory. The usual notation is multiplicative. That is, group operation is usually denoted by juxtaposition without a dot: ab
rather than a ∗ b, and even called multiplication. There are obvious notations an = aa . . . a
(n factors), a−1 for the inverse of a, a−n = (a−1 )n for n ∈ N. The identity is often denoted
by 1 in the group theory, but we will in general situations continue to use e to denote the
identity element of a group. We shall agree to write a0 = e for any element a in the group.
2.2
Further examples of groups
Example 2.5. For each n ∈ N, the n complex solutions to the equation xn = 1 (‘the nth
roots of unity’) form a multiplicative Abelian group Un . To prove this, let us first check
that Un is closed with respect to the multiplication. Indeed, xn = 1 and y n = 1 implies
(xy)n = xn y n = 1 × 1 = 1. Next, clearly, multiplication is associative and commutative
operation in Un ; 1 is the identity element. Finally, for each x ∈ Un , x−1 ∈ Un (for (x−1 )n =
(xn )−1 = 1−1 = 1), thus each element in Un has an inverse. Explicitly, Un consists of the
points ei(2π)k/n , k = 0, 1, . . . n − 1.
Example 2.6. Fix a natural number n. Consider the functions α : {1, . . . , n} → {1, . . . , n}.
We shall use the following notation for these functions. Suppose that n = 4 and α(1) = 3,
α(2) = 2, α(3) = 4, α(4) = 1. Then we write α as a 2 × 4 table
1 2 3 4
α=
.
(2.3)
3 2 4 1
In general, we write α as a 2 × n table, the entries of the top row are 1, 2, . . . , n, and
the entry under k is α(k). Let us restrict our attention to the functions α such that in
the bottom row of the table for α, each number k ∈ {1, . . . , n} appears once and only
once. Such functions are called permutations on a set {1, . . . , n}. Let us denote the set
of all permutations on {1, . . . , n} by Sn . The multiplication of permutations is defined as
composition of functions: αβ(k) = α(β(k)), k = 1, . . . , n. For example, let µ1 , µ2 ∈ S3 be
given by
1 2 3
1 2 3
,
µ2 =
.
µ1 =
1 3 2
3 2 1
Then
µ 1 µ2 =
1 2 3
,
2 3 1
µ2 µ 1 =
7
1 2 3
.
3 1 2
We have already seen that composition of functions is associative. The identity element in
Sn is the identity map:
1 2 ... n
ι=
.
1 2 ... n
The inverse of an element is the inverse function; e.g. for an α given by (2.3), the inverse is
1 2 3 4
−1
.
α =
4 2 1 3
One easily checks that Sn forms a group. This group is called a symmetric group on n
elements. Later on we shall consider this group in more detail.
Note that in the above example µ1 µ2 = µ2 µ2 , so S3 is not commutative.
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