4 Subgroups, Subrings and Subfields

Given a group, ring or field, it is natural to ask when a given subset of that algebraic structure obeys the
same axioms or inherits the same types of properties as the parent structure.

For example, the rationals form a field contained in the larger field of real numbers. The integers form a
ring, but not a field even though they are contained in the field Q. We say that Q is a subfield of R and that
Z is a subring of Q.
Definition 4.1. Let G be a group wrt the operation ∗ and let H be a non-empty subset of G. We say that
H is a subgroup of G, and write H < G, if H itself is a group wrt ∗.
We can define analogous substructures corresponding to rings and fields.
Definition 4.2. Let R be a ring and let S be a non-empty subset of R. We say that S is a subring of R if
S is itself a ring wrt the same operations as R.
Definition 4.3. Let F be a field and let K be a non-empty subset of F . We say that K is a subring of F
if K is itself a field wrt the same operations as F .

If S is a subring of a ring R we write S < R, and if K is a subfield of a field K we write F < K, where the
interpretation of < as ’subgroup’, ’subring’, or ’subfield’ is clear from the context.

Checking through all axioms of a group/ring/field can be time-consuming. The following results make this
much easier.
Lemma 4.1. Let G be a group wrt the operation ∗ and let H be a non-empty subset of G. Then H is a
group wrt ∗ iff
1. a ∗ b ∈ H for all a, b ∈ H (i.e. ∗ is an operation on H),
2. a−1 ∈ H for every a ∈ H.
Proof. If 1 holds then ∗ is an operation on H. It must also be assoc on H since if the assoc law holds in
G then it must certainly hold for any subset of G. Both 1 and 2 imply that aa−1 = e ∈ H for any a ∈ H,
where e is the identity element of G wrt ∗, so all three axioms of a group are satisfied. The converse is
immediate.
Example 4.1. The set C4 is a subgroup of C12 wrt complex multiplication. If z ∈ C4 then z 4 = 1, so
z 12 = 13 = 1 and hence z 4 is also a complex 12th root of unity. It follows that C4 ⊂ C12 . If z, w ∈ C4 then
(zw)4 = z 4 w4 = 1 · 1 = 1, so zw ∈ C4 . Finally, if z ∈ C4 then zz 3 = z 4 = 1 and (z 3 )4 = (z 4 )3 = 13 = 1, so
z −1 = z 3 ∈ C4 . It follows that C4 < C12 .
Example 4.2. The set   
a b
U T GL2(Q) = : a, b, c ∈ Q, a, c 6= 0
0 c
is a subgroup of the group of invertible 2 × 2 matrices with coefficients in Q, GL2 (Q). It is clearly a subset,
the product of any pair of invertible upper triangular matrices is also invertible and upper triangular, and
the inverse of any matrix in U T GL2 (Q) can be computed as
 −1  
a b c −b
= 1/ac .
0 c 0 a

We can apply similar techniques to obtain criteria for determining exactly when a subset of a ring (resp.
field) is a subring (resp. subfield).
Lemma 4.2. Let R be a ring and let S be a non-empty subset of R. Then S is a subring of R iff
1. a + b ∈ S for all a, b ∈ S (i.e. + is an operation on S),

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 2. ab ∈ S for all a, b ∈ S (i.e. · is an operation on S),
3. −a ∈ S for every a ∈ S.
Moreover, if R is a field then S is a subfield of R iff 1,2,3 hold and
4. a−1 ∈ S for every a ∈ S.
√ √ √ √
Example √ 4.3. Consider the set Z( 2) = {a + 2b : a, b ∈ Z}. Clearly Z( 2) ⊂ R. Let α = a + 2b and
β = c + 2d for some integers a, b, c, d. Then
√ √ √
α + β = (a + c) + 2(b + d), αβ = (ac + 2bd) + 2(ad + bc) and − α = −a + 2(−b),
√ √ √
so from Lemma 4.2, Z( 2) is a subring of R. If α−1 ∈ Z( 2) for any nonzero α in √ Z( 2) then we’ll have
shown that it is in fact a subfield of the field of real numbers. Suppose
√ that α = a + 2b =
6 0. Now α−1 ∈ R,
since R is a field. We need to check that it does in fact lie in Z( 2). Computing the inverse of α, we find
that   √ ! √
−1 1 1 a − 2b a − 2b
α = √ = √ √ = 2 .
a + 2b a + 2b a − 2b a − 2b2

It is easy to show that a2 6= 2b2 for any integers a, b, not both zero. However, note that
a −b
, 2
a2 − 2b a − 2b2
2

√ √
are not always integers for any a, b ∈ Z. This means that arbitrary α−1 ∈
/ Z( 2), so Z( 2) is not a subfield
of R. On the other hand, we can say that
a −b
, ∈Q
a2 − 2b2 a2 − 2b2
for√any pair of √
integers a, b, not both zero. In fact, applying a similar argument to the above we obtain that
Q( 2) = {a + 2b : a, b ∈ Q} is a subfield of R.
The following gives an example of a division ring, i.e. a unital ring in which every non-zero element has an
inverse, but where multiplication is not necessarily commutative.
Example 4.4. Let Q be the subset of all 2 × 2 complex matrices defined by
  
z w
Q= : w, z ∈ C .
−w̄ z̄

We claim that Q is a subring of M2 (C). Let

   
z w x y
p= and q = ,
−w̄ z̄ −ȳ x̄

for some complex numbers z, w, x, y. Then

   
z+x w+y z+x w+y
p+q = = ∈ Q,
−w̄ − ȳ z̄ + x̄ −w + y z+x

and    
zx − wȳ zy + wx̄ zx − wȳ zy + wx̄
pq = = ∈ Q,
−w̄x − z̄ ȳ −w̄y + z̄ x̄ −zx − wȳ zy + wx̄
using Lemma 3.1 (the additive and multiplicative properties of complex conjugation), so matrix addition and
multiplication are operations on Q. It is also easy to see that
 
−z −w
−p = ∈ Q,
w̄ −z̄

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so Q is indeed a subring of M2 (C). An interesting property of Q is the following:
      
z w z̄ −w z z̄ + ww̄ 0 1 0
= = (z z̄ + ww̄) .
−w̄ z̄ w̄ z 0 z z̄ + ww̄ 0 1

Note that z z̄ + ww̄ = |z|2 + |w|2 is a positive real number as long as p is not the all-zero matrix. We get the
same result if we perform the product in the reverse direction, which means that p has an inverse in Q, and
this is given by  
−1 1 z̄ −w
p = .
(|z|2 + |w|2 ) w̄ z
At this stage it is tempting to think that Q is a field, until we remember that matrix multiplication is not
commutative, for example, the matrices
   
i 0 1 i
and
0 −i i 1

do not commute.
Q is close to satisfying the axioms of a field, the only property it lacks is commutativity of multiplication.
Rings of this type are called skew fields or division rings.

The matrix ring Q described above is actually the set of quaternions. This division ring was discovered by
Hamilton in 1843, and will be discussed in the next section.

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