

Ratio Mathematica Volume 39, 2020, pp. 137-145

On Homomorphisms from Cn to Cm

Vinod S*
Biju G.S†

Abstract

In this paper, using elementary algebra and analysis, we characterize
and compute all continuous homomorphism from Cn to Cm. Also
we prove that the cardinality of the set of all non-continous group
homomorphism from Cn to Cm is at least the cardinality of the con-
tinuum.
Keywords: homomorphism; continuous function; Hamel basis;
2010 AMS subject classifications: 97U99. 1

*Department of Mathematics, Government College for Women, Thiruvananthapuram, Kerala,
India; wenod76@gmail.com.

†Department of Mathematics, College of Engineering, Thiruvananthapuram-695016, Kerala,
India; gsbiju@cet.ac.in.

1Received on June 27th, 2020. Accepted on December 15th, 2020. Published on December
31st, 2020. doi: 10.23755/rm.v39i0.562. ISSN: 1592-7415. eISSN: 2282-8214. ©Vinod et al.
This paper is published under the CC-BY licence agreement.

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Vinod S, Biju G.S

1 Introduction

Hamel [1905] introduced the concept of basis for real numbers and proved its
existence in 1905 by exploring functions which satisfy Cauchy’s functional equa-
tion f(x + y) = f(x) + f(y) for all x,y ∈ R. Using the existence of such a
basis, he described all solutions of Cauchy’s functional equation and established
the existence of discontinuous solutions. Cauchy demonstrated that any additive
function is rationally homogeneous. He also proved that the only continuous addi-
tive functions are real homogeneous and thus linear, and that an additive function
with a discontinuity is discontinuous throughout. Further restrictions were placed
on a non-linear additive function by Darboux [1875] who showed in 1875 that an
additive function bounded above or below on some interval is continuous, hence
linear. A survey of the research concerning additive functions can be found in
Green and Gustin [1950]

The continuous ring homomorphisms from C to C are trivial map, identity
map and complex conjugation. Since C is a field, all non-trivial ring homomor-
phisms are automorphisms on C. Thus identity map and complex conjugation are
the only continuous automorphisms on C. Any automorphisms on C other than
identity and complex conjugation is called a ”wild” automorphism on C. Kestel-
man [1951] proved the existence of so-called wild automorphism on C and the
showed that the set of such automorphisms on C has cardinality 2c. Many prop-
erties of wild automorphism on C are still open.

Calculating the number of homomorphisms between two groups or two rings
is a fundamental problem in abstract algebra. It is not easy to determine the num-
ber of distinct homomorphism between any two given groups or rings. Most of
the current results in this area are limited to groups or specific types of rings. For
example, Chigira et al. [2000] studied the number of homomorphisms from a fi-
nite group to a general linear group over a finite field. In a later study Bate [2007]
furnished the upper and lower limits for the number of completely reducible ho-
momorphisms from a finite group Γ to general linear and unitary groups over
arbitrary finite fields and to orthogonal and symplectic groups over finite fields of
odd characteristics. Matei and Suciu [2005] discusses a method for calculating
the number of epimorphisms from a finitely presented group G to a finite solvable
group Γ. Further discussion on homomorphisms on certain finite groups can be
found in Mal’cev [1983], Riley [1971], Hyers and Rassias [1992], but the solu-
tion to the general problem is still elusive. Hence the purpose of the paper is to
characterize and compute all continuous group homomorphisms from Cn to Cm.

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On Homomorphisms from Cn to Cm

2 Notations and Basic Results
Most of the notations, functions and terms we mentioned in this paper can be

find in Jacobson [2013], Gallian [1994] and Kestelman [1951].
We can interpret Hamel’s concept as follows. The set R of real numbers is a

linear space over the field Q of rational numbers. This linear space has a basis.
Namely, there exists a subset H ⊂ R such that every non-zero x ∈ R can uniquely
be written as a linear combination of the elements of H with rational coefficients.
That is, there exist distinct elements h1,h2, . . . ,hk of H and non-zero rational
numbers wh1 (x),wh2 (x), . . . ,whk (x) such that

x =
k∑

i=1

whi (x)hi (1)

Thus for x ∈ R, by adding the terms of the form 0 · hj in the representation (1),
we can write

x =
∑
h∈H

wh(x)h (2)

where wh(x) ∈ Q and wh(x) = 0 for all h except for a finite number of values of
h. Hamel based his argument on Zermelo’s fundamental result which states that
every set can be well ordered. Hamel’s argument is valid for an arbitrary linear
space L 6= {0} over a field. For this reason, recently such a basis is called a
Hamel basis(see also Cohn and Cohn [1981], Halpern [1966], Jacobson [2013],
Kharazishvili [2017]).

If f : R → R is additive, then it is easy to derive

f(rx) = rf(x)

for every r ∈ Q and x ∈ R. Thus, if H ⊂ R is a Hamel basis and x is a real
number, we obtain

f(x) = f

(∑
h∈H

wh(x)h

)
=
∑
h∈H

f

(
wh(x)h

)
=
∑
h∈H

wh(x)f(h). (3)

Observing that the Hamel bases of a linear space L coincide with the maximal
linearly independent subsets of L the existence of a Hamel basis is established
with the aid of Zorn’s maximum principle.

Theorem 2.1. Let L be a vector space over the field F . Then L has a Hamel
basis.

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Vinod S, Biju G.S

Theorem 2.2. Any continuous function f : Cn → C which assume only rational
values is constant.

Halbeisen and Hungerbühler [2000] showed that in an infinite dimensional
Banach space, every Hamel base has the cardinality of the Banach space, which
is at least the cardinality of the continuum.

Theorem 2.3. If K ⊂ C is a field and E is a Banach space over K such that
dim(E) = ∞, then every Hamel base of E has cardinality |E|.

3 Homomorphisms from Cn to Cm

First we will characterize all continuous group homomorphisms from Cn to
Cm .

Theorem 3.1. The cardinality of the set of continuous group homomorphisms
from Cn to Cm is equal to the cardinality of the continuum.

Proof. Let φ : Cn → Cm be a continuous group homomorphism. For
1 ≤ j ≤ n; denote ej for the n-tuple whose jth component is 1 and 0’s elsewhere,
and denote êj for the n-tuple whose jth component is i and 0’s elsewhere.

We will complete the proof by the following steps.

Step 1: φ(nej) = nφ(ej) and φ(nêj) = nφ(êj) for all n ∈ Z and for all
j (1 ≤ j ≤ n).

For n ∈ N, the argument is clear since φ is a group homomorphism.

Since φ is a group homomorphis,

φ(−nej) = −φ(nej) = −nφ(ej) and φ(0ej) = 0φ(ej)

Therefore φ(nej) = nφ(ej) for all n ∈ Z and for all j (1 ≤ j ≤ n). Similarly
we can prove φ(nêj) = nφ(êj) for all n ∈ Z and for all j (1 ≤ j ≤ n).

Step 2: φ(rej) = rφ(ej) and φ(rêj) = rφ(êj) for all r ∈ Q and for all
j (1 ≤ j ≤ n).

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On Homomorphisms from Cn to Cm

Let r =
p

q
, where p ∈ Z, q ∈ N. Then rq = p and hence rqej = pej. So

φ(rqej) = φ(pej)

=⇒ qφ(rej) = pφ(ej)

=⇒ φ(rej) =
p

q
φ(ej)

=⇒ φ(rej) = rφ(ej), for all r ∈ Q and for all j (1 ≤ j ≤ n).

Similarly, φ(rêj) = rφ(êj) for all r ∈ Q and for all j (1 ≤ j ≤ n).

Step 3: φ(xej) = xφ(ej) and φ(xêj) = xφ(êj) for all x ∈ R and for all
j (1 ≤ j ≤ n).

Let x ∈ R and 1 ≤ j ≤ n. Then there is a sequence (rm) of rational numbers
such that rm → x in R. Then rmej → xej as m → ∞. Since φ is continuous at
xej, we have

φ(xej) = lim
m→∞

φ(rmej)

= ( lim
m→∞

rm)φ(ej) ; by step 2

= xφ(ej)

Similarly, φ(xêj) = xφ(êj) for all x ∈ R and for all j (1 ≤ j ≤ n).

Step 4: Characterization of continuous homomorphisms from Cn to Cm.

Let z = (z1,z2, . . . ,zn) ∈ Cn. For 1 ≤ j ≤ n, let xj = Re(zj) and yj =
Im(zj). Then

z = (x1,x2, . . . ,xn) + (iy1, iy2, . . . , iyn)

=
n∑

j=1

xjej +
n∑

j=1

yjêj

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Vinod S, Biju G.S

So

φ(z) = φ

( n∑
j=1

xjej +
n∑

j=1

yjêj

)

=
n∑

j=1

φ(xjej) +
n∑

j=1

φ(yjêj)

=
n∑

j=1

xjφ(ej) +
n∑

j=1

yjφ(êj)

=
n∑

j=1

Re(zj)φ(ej) +
n∑

j=1

Im(zj)φ(êj).

Conversly, if aj(1 ≤ j ≤ n) and bj(1 ≤ j ≤ n) be 2n complex numbers , then
the map φ given by

φ(z1,z2, . . . ,zn) =
n∑

j=1

Re(zj)aj +
n∑

j=1

Im(zj)bj

is a continuous group homomorphism from Cn to Cm. Hence the cardinality of the
set of continuous group homomorphisms from Cn to Cm is same as the cardinality
of C2nm, which is the cardinality of the continuum. 2

Now, we provide a proof to the existence of non-continuous group homomor-
phism from Cn to Cm.

Theorem 3.2. The cardinality of the set of all non-continous group homomor-
phism from Cn to Cm is at least the cardinality of the continuum.

Proof. Consider Cn as a vector space over the field Q of rational numbers
and H be a Hamel basis of Cn over Q. Then every vector z ∈ Cn can be uniquely
expressed

z =
∑
h∈H

wh(z)h (4)

where wh(z) ∈ Q and wh(x) = 0 for all h except for a finite number of values of
h. Let e0 and ê0 are the zero elements in Cn and Cm respectively. Let e1 and ê1 are
the n−tuple and m−tuple respectively such that first component is 1 and all other
components are 0. Let h′ be a fixed element in H. Define a map ψh′ : Cn → Cm
by

ψh′ (z) = ψh′

(∑
h∈H

wh(z)h

)
= wh′ (z)ê1.

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On Homomorphisms from Cn to Cm

Let z =
∑
h∈H

wh(z)h and z′ =
∑
h∈H

wh(z
′)h be two elements in Cn. Then

ψh′ (z + z
′) = ψh′

(∑
h∈H

wh(z)h +
∑
h∈H

wh(z
′)h

)
= ψh′

(∑
h∈H

[wh(z) + wh(z
′)]h

)
= [wh′ (z) + wh′ (z

′)]ê1

= wh′ (z)ê1 + wh′ (z
′)ê1

= ψh′ (z) + ψh′ (z
′).

Hence ψh′ : Cn → Cm is a group homomorphism.
For z = (z1,z2, . . . ,zm) ∈ Cm, define φ : Cm → C by φ(z) = z1. Then φ is

a continuous function. Define g : Cn → C by g(z) = φ◦ψh′ (z) for all z ∈ Cn.
Then for z =

∑
h∈H

wh(z)h ∈ Cn,

g(z) = φ◦ψh′
(∑

h∈H

wh(z)h

)
= φ(wh′ (z)ê1) = wh′ (z) ∈ Q,

g(h′) = g

(
1 ·h′ +

∑
h∈H,h 6=h′

0h

)
= φ◦ψh′

(
1 ·h′ +

∑
h∈H,h 6=h′

0h

)
= φ(1 · ê1) = 1

and

g(e0) = φ◦ψh′ (e0) = φ◦ψh′
(

0 ·h′ +
∑

h∈H,h 6=h′
0h

)
= φ(0 · ê1) = 0.

Hence g is a non-constant function from Cn to C which assumes only rational
values. Therefore g is not continuous and which gives the function ψh′ is discon-
tinuous.

Let h′ and h′′ be two distinct elements in H. Then

ψh′ (h
′) = ψh′ (1 ·h′) = 1 · ê1 = ê1

and

ψh′′ (h
′) = ψh′′ (0 ·h′′ + 1 ·h′) = 0 · ê1 = ê0.

Therefore ψh′ and ψh′′ are distinct. Then the cardinality of set of all non-continuous
group homomorphism from Cn to Cm is at least |H| = |Cn| = the cardinality of
the continuum. 2

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Vinod S, Biju G.S

4 Conclusions
In this paper, we characterized all continuous group homomorphisms from

Cn to Cm . Also we proved that the cardinality of the set of all non-continous
group homomorphism from Cn to Cm is at least the cardinality of the continuum.

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