ple, 3 5 5 3. The same cannot be said of shuff les. If shuff le g simply
f lip-f lops the top two cards (and leaves the other cards in the same posi-
tion), and if shuff le h just f lip-f lops the second and third card, let’s follow
what happens to the third card in the deck. If we perform g first, the third
card stays where it is, but then migrates to position 2 after we then per-
form h. If we perform h first, the third card initially moves to position 2,
and then g moves it to position 1. So performing the shuff les in different
orders produce different results—the order of shuff ling (multiplication
in this group) does make a difference.
Although a standard deck of cards contains fifty-two cards, one could
obviously shuff le a deck of any number of cards. The group of all pos-
sible shuff les of a deck of n cards is known as the symmetric group Sn.
The structure of S n —that is, the number and characteristics of its sub-
groups—becomes more complex the higher the value of n, and this is the
key fact that determines why the quintic has no solution in terms of radi-
cals.
Niels Henrik Abel (1802–1829)
Niels Henrik Abel was born into a large, and poor, Norwegian family. At
the age of sixteen, he embarked upon a program of reading the great
works of mathematics; but when he was eighteen, his father died. Abel,
though not in good health himself, assumed the responsibility for taking
care of his family. Despite these obligations, he decided to attack the
quintic, and initially thought he had obtained a solution in the manner of
Cardano and Ferrari. After realizing that his proof was in error, he came
to precisely the opposite conclusion: it was impossible to find an algebraic
expression for the roots of the general quintic. Working along the same
general lines as Ruffini, but avoiding the proof pitfalls that had plagued
the Italian mathematician, Abel was able to show that the general quintic
could not be solved by radicals, bringing to an end a quest that had started
more than three millennia earlier in Egypt.
After publishing a memoir outlining his proof, Abel went to Berlin,
where he began publishing his results on a variety of topics in the newly
launched Crelle’s Journal. These results were favorably viewed by German
mathematicians, and Abel then traveled to Paris, where he hoped to ob-
tain recognition from the leading French mathematicians.
However, France was a hotbed of mathematical activity, and Abel wrote
to a friend, “Every beginner has a great deal of difficulty getting noticed
here.”^10 Discouraged and weakened by tuberculosis, Abel returned home,
where he died at the tragically young age of twenty-seven. Unbeknownst
The Hope Diamond of Mathematics 93