If $n\geq 5$ prove that $A_n$ is the only nontrivial normal subgroup in $S_n$. But please don't use the fact $A_n$ is simple if $n\geq 6$.

Question

I am reading "Topics in Algebra 2nd Edition" by I. N. Herstein.
The following problem is Problem 17 on p.81 in this book.

Problem 17:
If $n\geq 5$ prove that $A_n$ is the only nontrivial normal subgroup in $S_n$.

I was able to understand the following answer to this problem if I assume $A_6,A_7,A_8,\dots$ are all simple:

There is no proof of the following fact before p.81 in this book:

$A_6,A_7,A_8,\dots$ are all simple.

And the following problem (Problem 14) is on p.81:

Problem 14:
Prove that $A_5$ has no normal subgroups $N\neq (e), A_5$.

I solved Problem 14. But I had to use several facts the author didn't write before p.81.

Is there any solution which doesn't use the fact $A_6,A_7,A_8,\dots$ are all simple?

Or does the author expect the reader proves the fact $A_6,A_7,A_8,\dots$ are all simple?

Yes, see Lemma $2.4$ in K. Conrad's notes here. It doesn't use that $A_n$ is simple for $n\ge 5$. — Dietrich Burde, Feb 18 '22 at 15:01
@DietrichBurde Thank you very much for your link. I am not sure if I can understand the notes, but I will try to read the notes. Thank you. — tchappy ha, Feb 18 '22 at 15:03
I am sure you can understand the notes, and in particular Lemma 2.1 or 2.4. It is written in all detail and is very elementary. — Dietrich Burde, Feb 18 '22 at 15:05
@DietrichBurde As you said, K. Conrad's notes were written in all detail and were very elementary. And the notes were self-contained. I was able to understand Lemma 2.4. Thank you very much! — tchappy ha, Feb 19 '22 at 03:15

0 Answers0