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Koro
Koro
08:17
Let $g$ be a unit in $Z_n$, let g be of order m. $\sigma_g: x\to x.i \pmod n$ is a permutation. How can I find the cyclic decomposition of $\sigma_g$?
For example: if we take Z_5, the $\sigma_2=(1243)$
In $Z_6, \sigma_5=(15)(24)$
But I'm not sure how to do it for the general case.
Martin Sleziak
Martin Sleziak
@Koro Did you mean $x\mapsto x\cdot g$ rather than $x\mapsto x\cdot i$?
Each cycle will be of the form $(a,ag,\dots,ag^{m-1}$, right?
I meant: $(a,ag,\dots,ag^{m-1})$. (Somehow I missed the right bracket.)
 
4 hours later…
Koro
Koro
12:31
Indeed, i=g. Sorry for the typo.
Martin Sleziak
Martin Sleziak
in Mathematics, 4 hours ago, by Koro
I’ll think about existence of T that you mentioned. I am not sure why that will exist considering that Z* may not be a group.
in Mathematics, 1 min ago, by Martin Sleziak
@Koro $(\mathbb Z_n^\times,\cdot)$ (the set of all units) is a group.
in Mathematics, 49 secs ago, by Martin Sleziak
And you also know that if $\gcd(a,n)=1$ then $a^{\varphi(n)} \equiv 1 \pmod n$; from Euler's theorem.
← 3 messages moved from Linear & Abstract algebra

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