Algebra , Assignment 8
Solutions
1. Three problems about manipulating products of cyclic groups.
(a) Write
Z
2
×
Z
2
×
Z
2
×
Z
9
×
Z
5
×
Z
25
as the product of cyclic
groups
Z
a
i
, 1
≤
i
≤
s
(you find the
s
) with
a
i
dividing
a
i
+1
for
all 1
≤
i < s
.
Solution:
We line up each power separately right justified:
Z
2
Z
2
Z
2


Z
9

Z
5
Z
25
Multiplying down the columns (note the numbers are relatively
prime in the column so this can be done) gives the answer
Z
2
×
Z
10
×
Z
450
.
(b) Write
Z
4
×
Z
40
×
Z
200
×
Z
1400
as the product of cyclic groups of
prime power order. (Prime power includes primes themselves.)
Solution:
Break each up into its prime powers giving
Z
4
×
Z
8
×
Z
5
×
Z
8
×
Z
25
×
Z
8
×
Z
25
×
Z
7
(c) Write
Z
5
×
Z
6
×
Z
7
in
both
of the above forms.
Solution:
For the second we have
Z
5
×
Z
2
×
Z
3
×
Z
7
. For the
first each prime power appears only once so we have
Z
210
.
2. Let
G
=
Z
5
×
Z
25
×
Z
125
. Find the order of (3
,
10
,
12). Give a good
description and a precise count on those (
a,b,c
)
∈
G
order 25.
Solution:
i
(3
,
10
,
12) = (0
,
0
,
0) requires 5

i
, 5

i
and 125

i
so it need
125

i
so the order is 125.
(The order must be a power of 5 as
o
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 Fall '08
 JOSEPHS
 Algebra, Prime number

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