, and for reference purposes the tables
for both 
and are given
below.
Class members had no trouble completing the multiplication
table for , and for reference purposes the tables
for both and are given
below.
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The second challenge issued for this day proved to be a
difficult one. It was: determine, if possible, which numbers in
correspond to which numbers in . It was obvious that 0 in corresponded
to 1 in since each acts as an identity. Another observation was that 3 in
and 6 in were the only elements whose squares equal the respective
identities, thus these two numbers appear to correspond. (In this instance
we are a bit loose with the language. We are considering 3 + 3 in
to be "3 squared," although that is not technically the same as
the usual definition of "squared" in the real number system.)
Beyond that, it was not obvious by looking at the two tables which of the
other four numbers in acted like which four numbers in ,
if indeed that is the case.
It suggested to the class that since 3 in and 6 in
were comparable by their squares (powers of two) equaling the respective
identities, then examining higher powers of the other elements might provide
similar information. Indeed:
| In group : |
In group : |
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These comparisons of powers are in fact an examination
of the orders of each element. The order of an element is the smallest power
of that element greater than one which results in the identity. (The identity
is an exception, being considered to have an order of one.) In symbols,
k is the order of x if 
identity
and k is the smallest integer larger than 1, and this is represented
as 
.
Each group and matches up exactly in terms
of numbers of elements of each order. Specifically, there are 1, 1, 2 and
2 elements of order 1, 2, 3 and 6, respectively. This observation gives
these groups a quality not unlike a DNA fingerprint. Are there other mathematical
groups with the same order structure as these two?
An example was given using these complex numbers:
. These six numbers form the set of all sixth roots of
1, or the complex number solutions to the algebra equation 
. Under ordinary complex number multiplication it can be seen that
these numbers form a similar group to and . They also, when
considered as points on the unit circle in the complex plane (equally spaced
points at intervals of 60 degrees, that is), form yet another mathematical
group if you consider rotation of the points along the circle to be the
binary operation.
Thus we have at least four different descriptions of what
could be considered otherwise similar groups. All these enjoy elements with
the same orders and the group binary operation is commutative. (Commutative
means that for any two elements a and b in a group, 
.
The property of commutivity, which is not present in all mathematical groups,
makes such groups "abelian," a term in honor of the mathematician
Neils Abel of Norway who died around 1820 at the age of 21.)
The notion of isomorphism was introduced at this point. Informally, two groups are isomorphic if they identical in behavior, i.e., basically the same except for symbols and the binary operation. Formally, two groups are isomorphic if the following holds:
. (That assumes that "addition" is the binary
operation in the first group and "multiplication" is the binary
operation in the second group.)
When the above properties hold, then f (x) is called an isomorphism between the two groups.
We chose to correspond elements by order, but since there
were two elements each of order three and six, it was unclear how to match
elements up. We simply used the arrangement above and declared the following
one to one correspondance between and :
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Thus we can define a function f (x) mapping
elements in to by following the table above; i.e.,
f (0) = 1, f (1) = 3, etc. Verifying that f (x)
indeed is an isomorphism is tedious at this stage, but the author later
constructed the following color coded tables using this one to one correspondance:
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It is apparent that by rearranging the elements in
to match their counterparts in that each group has identical properties
since the table patterns are identical. It should be clear that the homomorphism
property is satisfied since an addition in lands in the
same position on the table as the corresponding multiplication in .
The author and his colleague, George Carver, later worked
out a polynomial form of this f (x). Fitting the six pairs
of numbers to fifth degree polynomials, then reducing all coefficients to
modulo 7 produced this function: 
.
There are actually three other ways to match elements between these two groups by orders of the elements. One of those ways produced another table match similar to above, but the other two produced different tables by color and in fact the function pairing the elements together in each case was not a true isomorphism. This raises interesting issues worth investigating.
Returning at last to the big picture of how to determine
the length of repetends of 1 ÷ p given a prime p, the
class was informed that in the branch of mathematics known as group theory
one may prove that the multiplicative group 
is always isomorphic
to the additive group 
. Since the arithmetic in is somewhat easier
than that in , if we knew what 10 corresponded to in then it may be
much easier to determine its order in . This process amounts to
finding the isomorphism between the two groups.
Is there a general method for finding the isomorphism between
and ? We (Davidson and Carver) are unaware if this has been
accomplished. It would seem to be a significant discovery which could be
useful to number theory.
The class was left with these two challenges:
?