1

If $f$ is an automorphism of $S_3$, prove that there exists $\sigma\in S_3$ such that $f(\tau)=\sigma\tau\sigma^{-1}$ for every $\tau\in S_3$.

I am not sure what the question is asking. I know that a 3-cycle can be generated from two 2-cycles. So we only need to consider the three transpositions. So am I suppose to consider what $f(\tau)$ equals to by plugging into $\tau=$ $(12), (13), (23)$, and for each $f(\tau)$, I am suppose to look for a specific $\sigma$ and then go through it for each $\tau$.
.
I know that $(13)(12)(31)=(23)$ and $(12)(13)(12)=(23)$. I also done the following calculations:

$(23)(12)(23)=(13)$
$(13)(12)(31)=(23)$
$(12)(12)(12)=(12)$
$(123)(12)(132)=(23)$
$(132)(12)(123)=(13)$


$(13)(23)(31)=(12)$
$(12)(23)(21)=(13)$
$(23)(23)(23)=(23)$
$(123)(23)(132)=(13)$
$(132)(23)(123)=(12)$


$(12)(13)(12)=(23)$
$(23)(13)(22)=(12)$
$(13)(13)(13)=(13)$
$(123)(13)(132)=(12)$
$(132)(12)(123)=(23)$


$(132)(123)(123)=(123)$
$(123)(123)(132)=(123)$
$(12)(123)(21)=(132)$
$(23)(123)(32)=(132)$
$(13)(123)(31)=(132)$


$(132)(132)(123)=(132)$
$(123)(132)(132)=(132)$
$(12)(132)(21)=(123)$
$(13)(132)(31)=(123)$
$(23)(132)(32)=(123)$


Can someone clarify the question for me please. Thank you in advance

Edit: Since the mods chose to close the question, I would like to give an example to see if this is what the question is asking. Is the question asking for me to do the following: for $\tau=(12)$, we have $f(12)=$ either $(12), (13)$ or $(23)$. If it is $\tau=(12)$, then $\sigma=(12)$ if $f(12)=(13)$, then $\sigma=(23)$ and if $f(12)=(23)$, then $\sigma=(13)$. If the example is not sufficient, what else am I asked to do in the problem. I am really confused over the wording of the question.

Seth
  • 3,325
  • To accompany the duplicate, see here. – Shaun Jan 30 '22 at 02:08
  • 1
    @Shaun I edited the question. I should say the book i used, at where I am in the text, the book did not talked much about inner automorphism other than it's definition in an exercises. The text I used introduced advanced concepts in earlier exercises and talks about them in later sections of later chapters. Also, I am not even sure by doing the question, what do I ended up showing as a result. – Seth Jan 30 '22 at 02:17
  • 1
    @Shaun Thank you for reopening the question. :) – Seth Jan 30 '22 at 02:19
  • It might help to note that conjugation of $\alpha\in S_n$ by $\beta\in S_n$ preserves the cycle structure of $\alpha$. – Shaun Jan 30 '22 at 02:20
  • You're welcome :) – Shaun Jan 30 '22 at 02:20
  • 1
    @Shaun the text at this point have not talked about conjugate yet. I am mainly going through Hungerford's abstract algebra 3rd edition text. In the previous edition of the text, the section on symmetric and alternating group comes after quotient groups, but in the 3rd edition, it moves right after the introduction to groups and right before it talks about cosets and normal subgroups, etc. So I feel like I am having to do the question with my hands tied. – Seth Jan 30 '22 at 02:23
  • 1
    @Shaun that is why I did all the calculations, because I am not sure what I am suppose to do by doing the question this way, but I am not sure if I am suppose to do it that way or what the point of the question is asking. – Seth Jan 30 '22 at 02:25

0 Answers0