The use of euler function to calculate the order of an element
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Just trying to check if I understand the material right. I would like to calculate $o(5)$ for $U_6$ (or $mathbb{Z}_6^times$). On one hand I think that we need to use the euler function to do so. But on the other hand we have the following theorem:
$$o(a)=min{ninmathbb{N}|a^n=e}$$
So as I understand, I need to find all the minimal $ninmathbb{N}$ so $5^n=1$ (although I'm not sure that $e=1$). From my previous thread I learned that $5^n=5+...+5,(mod,6)$. But there is no $ninmathbb{N}$ so $5^n=1$.
Also what will happen with bigger numbers? For example how to calculate $o(5)$ for $U_{27}$?
group-theory
asked 2 days ago
vesii
475
add a comment |
up vote
0
down vote
favorite
Just trying to check if I understand the material right. I would like to calculate $o(5)$ for $U_6$ (or $mathbb{Z}_6^times$). On one hand I think that we need to use the euler function to do so. But on the other hand we have the following theorem:
$$o(a)=min{ninmathbb{N}|a^n=e}$$
So as I understand, I need to find all the minimal $ninmathbb{N}$ so $5^n=1$ (although I'm not sure that $e=1$). From my previous thread I learned that $5^n=5+...+5,(mod,6)$. But there is no $ninmathbb{N}$ so $5^n=1$.
Also what will happen with bigger numbers? For example how to calculate $o(5)$ for $U_{27}$?
group-theory
asked 2 days ago
vesii
475
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Just trying to check if I understand the material right. I would like to calculate $o(5)$ for $U_6$ (or $mathbb{Z}_6^times$). On one hand I think that we need to use the euler function to do so. But on the other hand we have the following theorem:
$$o(a)=min{ninmathbb{N}|a^n=e}$$
So as I understand, I need to find all the minimal $ninmathbb{N}$ so $5^n=1$ (although I'm not sure that $e=1$). From my previous thread I learned that $5^n=5+...+5,(mod,6)$. But there is no $ninmathbb{N}$ so $5^n=1$.
Also what will happen with bigger numbers? For example how to calculate $o(5)$ for $U_{27}$?
group-theory
asked 2 days ago
vesii
475
Just trying to check if I understand the material right. I would like to calculate $o(5)$ for $U_6$ (or $mathbb{Z}_6^times$). On one hand I think that we need to use the euler function to do so. But on the other hand we have the following theorem:
$$o(a)=min{ninmathbb{N}|a^n=e}$$
So as I understand, I need to find all the minimal $ninmathbb{N}$ so $5^n=1$ (although I'm not sure that $e=1$). From my previous thread I learned that $5^n=5+...+5,(mod,6)$. But there is no $ninmathbb{N}$ so $5^n=1$.
Also what will happen with bigger numbers? For example how to calculate $o(5)$ for $U_{27}$?
group-theory
group-theory
asked 2 days ago
vesii
475
asked 2 days ago
vesii
475
asked 2 days ago
vesii
475
asked 2 days ago
vesii
475
asked 2 days ago
vesii
475
475
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
$5^n$ means repeating the group operation on $5$ $n$ times. Since we're dealing with the multiplicative group here, the group operation is multiplication, not addition.
answered 2 days ago
Y. Forman
11.3k423
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
$5^n$ means repeating the group operation on $5$ $n$ times. Since we're dealing with the multiplicative group here, the group operation is multiplication, not addition.
answered 2 days ago
Y. Forman
11.3k423
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
add a comment |
up vote
0
down vote
$5^n$ means repeating the group operation on $5$ $n$ times. Since we're dealing with the multiplicative group here, the group operation is multiplication, not addition.
answered 2 days ago
Y. Forman
11.3k423
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
add a comment |
up vote
0
down vote
up vote
0
down vote
$5^n$ means repeating the group operation on $5$ $n$ times. Since we're dealing with the multiplicative group here, the group operation is multiplication, not addition.
answered 2 days ago
Y. Forman
11.3k423
$5^n$ means repeating the group operation on $5$ $n$ times. Since we're dealing with the multiplicative group here, the group operation is multiplication, not addition.
answered 2 days ago
Y. Forman
11.3k423
answered 2 days ago
Y. Forman
11.3k423
answered 2 days ago
Y. Forman
11.3k423
answered 2 days ago
Y. Forman
11.3k423
11.3k423
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
add a comment |
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
Thanks for the fast replay. Although there is still no solution for $5^n=1$ right?
– vesii
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
@vesii There is. $5^2=25=1$ mod6
– Y. Forman
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
oh right! and how the euler equation comes in handy? can we calculate all of the numbers?
– vesii
2 days ago
add a comment |
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