- #1

jackmell

- 1,807

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Hi,

I'm studying Abstract Algebra and was wondering if someone could help me understand a formula for computing ##\big|\operatorname{aut} \mathbb{Z}_n^{*}\big|## using a formula described in this reference:http://www.msri.org/people/members/chillar/files/autabeliangrps.pdf

Here is the basic formula. I don't understand:

(1) the definitions for ##c_k## and ##d_k##,

(2) if the formula is referring to the same ##p## or the prime factors of ##n##, and

(3) how Sylow p-subgroups are related to this problem although I realize a finite abelian group is equal to a direct product of distinct Sylow p-subgroups as in the example below.

Given ##\mathbb{Z}_n^{*}=\mathbb{Z}_{p_1^{e_1}}\times \mathbb{Z}_{p_2^{e_2}}\times\cdots\times \mathbb{Z}_{p_n^{e_n}}##,

define the following ##2n## numbers:

##\begin{align*} d_k=\operatorname{max}\{l:e_l=e_k\}\\

c_k=\operatorname{min}\{l:e_l=e_k\}

\end{align*}

##

then:

##

\big|\operatorname{aut} \mathbb{Z}_n^{*}\big|=\prod_{k=1}^n p^{d_k}-p^{k-1}\prod_{j=1}^n \left(p^{e_j}\right)^{n-d_j}\prod_{i=1}^n \left(p^{e_i}-1\right)^{n-c_i+1}

##

For example, I know ##\mathbb{Z}_{100}^*\cong \mathbb{Z}_{2^2}^*\times\mathbb{Z}_{5^2}^*\cong\mathbb{Z}_2\times \mathbb{Z}_{20}## and that ##\big|\operatorname{aut}\mathbb{Z}_{100}^*\big|=32##. Could we perhaps go over the formula for this example? Also, I think this is also related to Sylow p-subgroups and I know the two Sylow p-subgroups for this group is ##S_2=\{1,7,43,49,51,57,93,99\}##, and ##S_5=\{1,21,41,61,81\}##.

I'd like to find the size of the first 500 and wrote a routine to find them using another way and I'd like to check my results with the formula above.

Ok thanks for reading,

Jack

I'm studying Abstract Algebra and was wondering if someone could help me understand a formula for computing ##\big|\operatorname{aut} \mathbb{Z}_n^{*}\big|## using a formula described in this reference:http://www.msri.org/people/members/chillar/files/autabeliangrps.pdf

Here is the basic formula. I don't understand:

(1) the definitions for ##c_k## and ##d_k##,

(2) if the formula is referring to the same ##p## or the prime factors of ##n##, and

(3) how Sylow p-subgroups are related to this problem although I realize a finite abelian group is equal to a direct product of distinct Sylow p-subgroups as in the example below.

Given ##\mathbb{Z}_n^{*}=\mathbb{Z}_{p_1^{e_1}}\times \mathbb{Z}_{p_2^{e_2}}\times\cdots\times \mathbb{Z}_{p_n^{e_n}}##,

define the following ##2n## numbers:

##\begin{align*} d_k=\operatorname{max}\{l:e_l=e_k\}\\

c_k=\operatorname{min}\{l:e_l=e_k\}

\end{align*}

##

then:

##

\big|\operatorname{aut} \mathbb{Z}_n^{*}\big|=\prod_{k=1}^n p^{d_k}-p^{k-1}\prod_{j=1}^n \left(p^{e_j}\right)^{n-d_j}\prod_{i=1}^n \left(p^{e_i}-1\right)^{n-c_i+1}

##

For example, I know ##\mathbb{Z}_{100}^*\cong \mathbb{Z}_{2^2}^*\times\mathbb{Z}_{5^2}^*\cong\mathbb{Z}_2\times \mathbb{Z}_{20}## and that ##\big|\operatorname{aut}\mathbb{Z}_{100}^*\big|=32##. Could we perhaps go over the formula for this example? Also, I think this is also related to Sylow p-subgroups and I know the two Sylow p-subgroups for this group is ##S_2=\{1,7,43,49,51,57,93,99\}##, and ##S_5=\{1,21,41,61,81\}##.

I'd like to find the size of the first 500 and wrote a routine to find them using another way and I'd like to check my results with the formula above.

Ok thanks for reading,

Jack

Last edited: