24
1. ALGEBRAIC THEMES
1.5.8.
Show that the multiplication in
S
n
is noncommutative for all
n
±
3
.
Hint:
Find a pair of 2–cycles that do not commute.
1.5.9.
Let
±
n
denote the perfect shufﬂe of a deck of
2n
cards. Regard
±
n
as a bijective function of the set
f
1;2;:::;2n
g
. Find a formula for
±
n
.j/
,
when
1
²
j
²
n
, and another formula for
±
n
.j/
, when
n
C
1
²
j
²
2n
.
1.5.10.
Explain why a cycle of length
k
has order
k
. Explain why the order
of a product of disjoint cycles is the least common multiple of the lengths
of the cycles. Use examples to clarify the phenomena for yourself and to
illustrate your explanation.
1.5.11.
Find the cycle decomposition for the perfect shufﬂe for decks of
size 2, 4, 6, 12, 14, 16, 52. What is the order of each of these shufﬂes?
1.5.12.
Find the inverse, in two–line notation, for the perfect shufﬂe for
decks of size 2, 4, 6, 8, 10, 12, 14, 16. Can you ﬁnd a rule describing the
inverse of the perfect shufﬂe in general?
The following two exercises supply important details for the proof of
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 Fall '08
 EVERAGE
 Algebra, Multiplication, perfect shuffle

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