The use of euler function to calculate the order of an element

Question

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: $$o(a)=min\{n\in\mathbb{N}|a^n=e\}$$ So as I understand, I need to find all the minimal $n\in\mathbb{N}$ 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 $n\in\mathbb{N}$ so $5^n=1$.

Also what will happen with bigger numbers? For example how to calculate $o(5)$ for $U_{27}$?

score 1 · Answer 1 · answered Nov 19 '18 at 18:14

1

$5^n$ 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 Nov 19 '18 at 18:14

BallBoy

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Thanks for the fast replay. Although there is still no solution for $5^n=1$ right? – vesii Nov 19 '18 at 18:18
@vesii There is. $5^2=25=1$ mod6 – BallBoy Nov 19 '18 at 18:19
oh right! and how the euler equation comes in handy? can we calculate all of the numbers? – vesii Nov 19 '18 at 18:22

The use of euler function to calculate the order of an element

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