72
1. ALGEBRAIC THEMES
Two groups
G
and
H
are said to be
isomorphic
if there is a
bijective
map
f
W
H
!
G
between them that makes the multiplication table of one
group match up with the multiplication table of the other. The map
f
is
called an
isomorphism
. (The requirement on
f
is this: Given
a;b;c
in
H
,
we have
c
D
ab
if and only if
f.c/
D
f.a/f.b/
.)
For another example, the permutation group
S
3
is isomorphic to the
group of symmetries of an equilateral triangular card, as is shown in Exer
cise
1.5.2
.
Another simple way in which two groups may be related is that one
may be
contained
in the other. For example, the set of symmetries of the
square card that do not exchange top and bottom is
f
e;r;r
2
;r
3
g
; this is
a group in its own right. So the group of symmetries of the square card
contains the group
f
e;r;r
2
;r
3
g
as a subgroup
. Another example: The set
of eight invertible 3by3 matrices
f
E;A;B;C;D;R;R
2
;R
3
g
introduced
in Section
1.4
is a group under the operation of matrix multiplication; so it
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 Fall '08
 EVERAGE
 Algebra, Multiplication, 3 g, multiplication table, inverses.

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