tive inverse x^1 that satisfies xx^1 x^1 x1. These are the key properties
used to define a group G, which is a collection of elements and a way of
combining two of those elements g and h into an element gh in G. This
way of combining elements is usually referred to as multiplication, and
the element gh that results is called a product, although as we shall see
there are many groups in which “multiplication” bears no resemblance to
arithmetic. The multiplication must satisfy the associative law: a(bc)(ab)c
for any three elements a, b, and c of the group. The group must contain an
identity element, which could be denoted by 1, which satisfies g 1 1 gg
for any member g of the group. Finally, each member g of the group must
have a multiplicative inverse g^1 , which satisfies gg^1 g^1 g1.
An interesting example of a group, which has an important and surpris-
ing connection to the problem of solving the quintic, is found by examin-
ing what happens when we shuff le a deck of cards. It is possible to
completely describe a shuff le by thinking of where cards end up relative
to where they start. For instance, in a perfect shuff le, the top twenty-six
cards are placed in the left hand and the bottom twenty-six cards in the
right. The mechanics of the classic “waterfall” shuff le releases the bot-
tom card from the right hand, then the bottom card from the left, then
the next-to-bottom card from the right hand, and so on, alternating cards
from each hand. We could describe the perfect shuff le by means of the
following diagram, which describes where a card starts in the deck and
where it ends up; the top card in the deck is in position 1, and the bottom
card in position 52.
Starting
Position 1 2 3... 24 25 26 27 28 29... 50 51 52
Ending
Position 1 3 5... 47 49 51 2 4 6... 48 50 52We could produce a shorthand for this using algebraic notation.Starting Position (x) Ending Position1 x 26 2 x 1
27 x 52 2 x 52
The set of all shuff les of a deck of cards forms a group. The product gh
of two shuff les g and h is the rearrangement that results from first per-
forming shuff le g, then shuff le h. The identity element of this group is
The Hope Diamond of Mathematics 91