a homomorphism is a function between two
groups that preserves the group
structure in each group it's a tool for
comparing two groups for similarities
sometimes two groups are more than just
similar they're identical in this case
we no longer call function a
homomorphism
instead we call it an isomorphism and we
say the two groups are isomorphic
let's give a precise definition of an
isomorphism
suppose we have two groups gnh and a
homomorphism f from g to h
recall that a homomorphism is a function
f such that f of x times y equals f of x
times f of y
now a homomorphism does not have to be
one to one
it does not have to be an injection
it's possible for many elements in g to
map to the same element in h
similarly f does not have to be onto it
does not have to be a surjection
but for the groups g and h to be
identical we need the homomorphism f to
be one to one and onto
f needs to be both an injection and
surjection
this way we can pair each element in g
with a unique element in h and vice
versa
this is a definition of an isomorphism
an isomorphism is a homomorphism that is
one to one and onto
this can be said more briefly as an
isomorphism is a homomorphism that is
also a bijection
let's now see some homomorphisms and
determine if they're isomorphisms
for our first example consider the
groups of the positive real numbers
under multiplication and all real
numbers under addition
one homomorphism between them is a
logarithm function we can check that
this is a homomorphism by using the laws
of logarithms
the log of x times y is equal to the log
of x plus the log of y
to see if this is an isomorphism we have
to test if it's a bijection let's first
check that this function is one to one
suppose that log of x equals log of y
since the logs are equal e to each power
is equal to
this simplifies to x equals y so this
function is one to one
next the range of the log function is
all real numbers so this function is
onto
so in this example the log function is a
homomorphism and a bijection that makes
it an isomorphism these two groups are
isomorphic
for our next example the first group
will be the non-zero complex numbers
under multiplication which we denote by
a c with a multiplication sign
here it's understood that you are not
including 0 because 0 does not have an
inverse under multiplication
the second group will be the complex
numbers with absolute value of 1 under
multiplication
we'll denote this group by s1
this is a standard notation when talking
about n-dimensional spheres the s is
short for sphere and the 1 tells us the
dimension
in this case s1 is just the circle in
the complex plane with a radius of 1.
recall that every complex number can be
written in polar form as r times e to
the i theta where r is the distance of
the complex number to the origin and
theta is how far you have to rotate from
the positive x axis to reach the complex
number
with this setup we can now define a
homomorphism between these two groups
the function is f of r times e to the i
theta equals e to the i theta
you can visualize this by taking any
non-zero complex number z
drawing away from the origin to z
and mapping z to the point where it
intersects s1
but is this a homomorphism
let's check
let z equal r times e to the i alpha and
w equal s times e to the i beta we want
to check that f of z times w equals f of
z times f of w
to begin substitute in the polar forms
next multiply the numbers on the left
using the definition of the function f
we can see what each value maps to
this gives us e to the i alpha plus i
beta equals e to the i alpha times e to
the i beta
this is true by the rules of exponents
so this function is indeed a
homomorphism
it is also on too because for every
point in the circle there are an
infinite number of complex numbers which
map to it
but for the same reason this function is
not one to one so while this is a
homomorphism it is not an isomorphism
let's recap an isomorphism is a
homomorphism that is also bijection
if you have an isomorphism between two
groups then you say the groups are
isomorphic
this means they have the exact same
group structure even if they look
different from each other
the word isomorphism reflects the
definition iso means equal and morph
means shape
i doubt anyone could think of a better
name than this
you