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70
1. ALGEBRAIC THEMES
Examples we have considered so far are as follows:
±
The set of symmetries of a geometric ﬁgure with composition of
symmetries as the product.
±
The set of permutations of a (ﬁnite) set, with composition of per
mutations as the product.
±
The set of integers with addition as the operation.
±
Z
n
with addition as the operation. Indeed, Proposition
1.7.7
,
parts (a), (b), and (c), says that addition in
Z
n
is associative and
has an identity
Œ0Ł
, and that all elements of
Z
n
have an additive
inverse.
±
KŒxŁ
with addition as the operation. In fact, Proposition
1.8.2
,
parts (a), (b), and (c), says that addition in
KŒxŁ
is associative,
that
0
is an identity element for addition, and that all elements of
KŒxŁ
have an additive inverse.
It is convenient and fruitful to make a
concept
out of the common char
acteristics of these several examples. So we make the following deﬁnition:
Deﬁnition 1.10.1.
A
group
is a (nonempty) set
G
with a product, denoted
here simply by juxtaposition, satisfying the following properties:
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Permutations

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