3.1. DIRECT PRODUCTS
151
Since the
Q
A
i
are mutually commuting subgroups, the subgroup gener
ated by any collection of them is their product:
h
Q
A
i
1
;
Q
A
i
2
: : : : ;
Q
A
i
s
i D
Q
A
i
1
Q
A
i
2
Q
A
i
s
:
In fact, the product is
f
.x
1
; x
2
; : : : ; x
n
/
2
P
W
x
j
D
e
if
j
62 f
i
1
; : : : ; i
s
gg
:
In particular,
Q
A
1
Q
A
2
Q
A
n
D
P
.
Example 3.1.11.
Suppose the local animal shelter houses several collec
tions of animals of different types: four African aardvarks, five Brazilian
bears, seven Canadian canaries, three Dalmatian dogs, and two Ethiopian
elephants, each collection in a different room of the shelter.
Let
A
denote the group of permutations of the aardvarks,
A
Š
S
4
.
1
Likewise, let
B
denote the group of permutations of the bears,
B
Š
S
5
; let
C
denote the group of permutations of the canaries,
C
Š
S
7
; let
D
denote
the group of permutations of the dogs,
D
Š
S
4
; and finally, let
E
denote
the group of permutations of the elephants,
E
Š
S
2
Š
Z
2
. The symmetry
group of the entire zoo is
P
D
A
B
C
D
E
.
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Permutations, Z5 Z3 Z2

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