Finding all of the element in a product group
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I would like to find all of the element in the following group: $U_ntimes mathbb Z_n$ when $U_n$ is the Multiplicative group of integers modulo $n$ and $mathbb{Z}_n$ is the Sum group of integers modulo.
For example, How can I find part or all of the element in $U_{10}timesmathbb{Z}_{10}$?
group-theory
asked Dec 1 '18 at 20:21
vesiivesii
1156
$endgroup$
add a comment |
$begingroup$
I would like to find all of the element in the following group: $U_ntimes mathbb Z_n$ when $U_n$ is the Multiplicative group of integers modulo $n$ and $mathbb{Z}_n$ is the Sum group of integers modulo.
For example, How can I find part or all of the element in $U_{10}timesmathbb{Z}_{10}$?
group-theory
asked Dec 1 '18 at 20:21
vesiivesii
1156
$endgroup$
$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06
add a comment |
$begingroup$
I would like to find all of the element in the following group: $U_ntimes mathbb Z_n$ when $U_n$ is the Multiplicative group of integers modulo $n$ and $mathbb{Z}_n$ is the Sum group of integers modulo.
For example, How can I find part or all of the element in $U_{10}timesmathbb{Z}_{10}$?
group-theory
asked Dec 1 '18 at 20:21
vesiivesii
1156
$endgroup$
I would like to find all of the element in the following group: $U_ntimes mathbb Z_n$ when $U_n$ is the Multiplicative group of integers modulo $n$ and $mathbb{Z}_n$ is the Sum group of integers modulo.
For example, How can I find part or all of the element in $U_{10}timesmathbb{Z}_{10}$?
group-theory
group-theory
asked Dec 1 '18 at 20:21
vesiivesii
1156
asked Dec 1 '18 at 20:21
vesiivesii
1156
asked Dec 1 '18 at 20:21
vesiivesii
1156
asked Dec 1 '18 at 20:21
vesiivesii
1156
asked Dec 1 '18 at 20:21
vesiivesii
1156
1156
$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06
add a comment |
$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06
$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06
add a comment |
1 Answer
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$begingroup$
$U_{10}oplus mathbb Z_{10}cong mathbb Z_{4}oplus mathbb Z_{10}$
Can you do now?
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
$endgroup$
add a comment |
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1 Answer
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$begingroup$
$U_{10}oplus mathbb Z_{10}cong mathbb Z_{4}oplus mathbb Z_{10}$
Can you do now?
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
$endgroup$
add a comment |
$begingroup$
$U_{10}oplus mathbb Z_{10}cong mathbb Z_{4}oplus mathbb Z_{10}$
Can you do now?
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
$endgroup$
add a comment |
$begingroup$
$U_{10}oplus mathbb Z_{10}cong mathbb Z_{4}oplus mathbb Z_{10}$
Can you do now?
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
$endgroup$
$U_{10}oplus mathbb Z_{10}cong mathbb Z_{4}oplus mathbb Z_{10}$
Can you do now?
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
answered Dec 2 '18 at 15:01
SRJSRJ
1,7431620
1,7431620
add a comment |
add a comment |
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$begingroup$
What is your definition of the direct product? Tuples with componentwise group operation or via the universal property?
$endgroup$
– Luke
Dec 1 '18 at 20:44
$begingroup$
@Luke thanks for the reply. The definition is $Gtimes H = {(g,h)|gin G,hin H}$
$endgroup$
– vesii
Dec 1 '18 at 21:05
$begingroup$
That is a set, and not yet a group. How is the multiplication defined?
$endgroup$
– Luke
Dec 1 '18 at 21:06