Form of an element of a normal subgroup of $A_n$

Asked Mar 12 '15 at 11:39

Active Mar 12 '15 at 11:39

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I want to show that $A_n$ is simple for $n\geq 5$.

For $n=5$ I have used the following criterion

Let $H$ be a normal subgroup of $A_5$ then $H$ can contain any one of the following

$a.$ a $5$ cycle of the form $(a,b,c,d,e)$

$b.$ a $3$ cycle of the form $(a,b,c)$

and $c.$ product of two disjoint $2$ cycles $(a,b)(c,d)$

in all the cases I am able to show that a $3$ cycle is in $H$ and thus $H=A_5$

Now I want to argue in the general case .How should I split the cases here; because suppose $H$ is a normal subgroup of $A_n;n>5$ then how will I know what form an element of $H$ will have .Please help I am totally confused

asked Mar 12 '15 at 11:39

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See Theorems 2, 3, and 4 in https://rutherglen.science.mq.edu.au/ccooper/Groups/CHAP11%20Soluble%20Groups.pdf – Gerry Myerson Mar 12 '15 at 11:49
Have you had a chance to follow-up on that link? – Gerry Myerson Mar 16 '15 at 01:56
yes sir,it has helped me a lot @GerryMyerson – Learnmore Mar 16 '15 at 05:40
Good! Maybe when you're ready, you can summarize, and post an answer to your question. – Gerry Myerson Mar 16 '15 at 06:08
Sir I have to appear in an interview regarding group theory .Can u please help which topics should I highlight more ;I cant figure out as the topic is so dense.@GerryMyerson – Learnmore Mar 16 '15 at 06:11
You had better speak to whomever it is who is going to interview you. I don't see how I could possibly advise you, when I know nothing about this interview. – Gerry Myerson Mar 16 '15 at 06:16

Form of an element of a normal subgroup of $A_n$

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