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Some useful facts I am trying to use:

If the multiplicative group $U_n$ modulo $n$ is a cyclic group, a generator $g$ of $U_n$ is called a primitive root of $n$.

if $g$ in $U_n$ is a primitive root, then $|g|= \phi(n)$ where $\phi$ is the euler phi function.

An element $g$ in $U_n$ is a primitive root if and only if $g^{\phi(n)/p}$ not congruent to $1\pmod n$ for each prime dividing $\phi(n)$

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But why is the title different from the content? –  awllower Mar 24 '13 at 5:52
    
Typesetting tip: since you are going to need dollars anyway (the character) to get math symbols, subscripts, etc., put them around the entire math formulas, then the whole thing will look better. –  Marc van Leeuwen Mar 24 '13 at 6:08
1  
This question and the one here: math.stackexchange.com/questions/339317 are the same - Presumably, two people from the same class, given the timing. –  Glen O Mar 24 '13 at 6:11
    
There is an inconsistency in the question in the title: $g$ and $b$ are elements in a cyclic group, yet their product figures in a congruence relation, as if they were numbers. It could be that you identify $U_n$ with the cyclic group $(\Bbb Z/n\Bbb Z,{+})$, but then you should use addition, not multiplication (and the congruence would not be explicitly needed). I think you are just saying $b$ is the inverse of $g$; then you should omit the $\pmod n$ part. –  Marc van Leeuwen Mar 24 '13 at 6:14
    
@GlenO Not merely a duplicate, the silly $\pmod n$ is present in the other question as well! –  Marc van Leeuwen Mar 24 '13 at 6:16

3 Answers 3

up vote 3 down vote accepted

Suppose that $g$ is a primitive root $\pmod n, \ g\cdot b\equiv1\pmod n$ and $b$ is not a primitive root $\pmod n$.

Then, use that $$ (x\cdot y)^k\equiv x^k\cdot y^k \pmod n $$ to derive a contradiction.

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Hint: if $\,C=\langle\,x\,\rangle\,$ is a cyclic group, then also $\,C=\langle\,x^{-1}\,\rangle\,$

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$$\text{ If }ord_ma=d, ord_m(a^k)=\frac{d}{(d,k)}$$ (Proof @Page#95)

Here $b\equiv g^{-1}\pmod n$

So, $k=-1\implies (ord_ng,-1)=1$

and as $g$ is a primitive root $\pmod n,ord_ng=\phi(n)$

$\implies ord_n(b)=ord_n(g^{-1})=\frac{ord_ng}1=ord_ng=\phi(n)$

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