- #1

jackmell

- 1,807

- 54

Hi,

I was wondering if anyone here could help me prove or disprove this empirical observation or explain why there seems to be a connection between automorphic generators described below and the automorphisms of these groups:

Consider the p-group expansion:

##\mathbb{Z}_n^*\cong S_2\times S_{p_2}\times\cdots\times S_{p_n}##.

Now define an automorphic generator ##G=\big<a_1,a_2,\cdots,a_n\big>## of ##\mathbb{Z}_n^*## as a generator meeting the following requirements:

## \displaystyle\bigcap^n \big<a_i\big> =\{1\}\quad \text{and} \quad \displaystyle \prod^n \big<a_i\big> =\mathbb{Z}_n^*

##

and consider the set ##A=\{\big<b_1,b_2,\cdots,b_n\big>\}## of all automorphic generators in which the generator elements match the orders of the elements of ##G##, that is ##o(a_1)=o(b_1)##, ##o(a_2)=o(b_2)## and so forth. Show or provide a contradiction:

##\begin{aligned}(1)\quad &\big|\operatorname{aut}\mathbb{Z}_n^*\big|=\big|A\big|,\\

(2)\quad &\operatorname{aut}\mathbb{Z}_n^*=\big<G,A\big>,\\

(3)\quad &\text{min}|G|=|S_2|

\end{aligned}

##

where we interpret ##\big<G,A\big>## as meaning any mapping of a generator ##G## to a set of generator mappings ##A##, and the expression ##\text{min}|G|=|S_2|## reflects the empirical observation that the minimum number of elements of an automorphic generator appears to be equal to the number of factors of ##S_2##.

Ok thanks,

Jack

I was wondering if anyone here could help me prove or disprove this empirical observation or explain why there seems to be a connection between automorphic generators described below and the automorphisms of these groups:

Consider the p-group expansion:

##\mathbb{Z}_n^*\cong S_2\times S_{p_2}\times\cdots\times S_{p_n}##.

Now define an automorphic generator ##G=\big<a_1,a_2,\cdots,a_n\big>## of ##\mathbb{Z}_n^*## as a generator meeting the following requirements:

## \displaystyle\bigcap^n \big<a_i\big> =\{1\}\quad \text{and} \quad \displaystyle \prod^n \big<a_i\big> =\mathbb{Z}_n^*

##

and consider the set ##A=\{\big<b_1,b_2,\cdots,b_n\big>\}## of all automorphic generators in which the generator elements match the orders of the elements of ##G##, that is ##o(a_1)=o(b_1)##, ##o(a_2)=o(b_2)## and so forth. Show or provide a contradiction:

##\begin{aligned}(1)\quad &\big|\operatorname{aut}\mathbb{Z}_n^*\big|=\big|A\big|,\\

(2)\quad &\operatorname{aut}\mathbb{Z}_n^*=\big<G,A\big>,\\

(3)\quad &\text{min}|G|=|S_2|

\end{aligned}

##

where we interpret ##\big<G,A\big>## as meaning any mapping of a generator ##G## to a set of generator mappings ##A##, and the expression ##\text{min}|G|=|S_2|## reflects the empirical observation that the minimum number of elements of an automorphic generator appears to be equal to the number of factors of ##S_2##.

Ok thanks,

Jack

Last edited: