Re: Normal subgroups

From: Stefan Fredriksson (mcriley_at_telia.com)
Date: 12/01/04 

Date: Wed, 01 Dec 2004 20:14:56 GMT

Arturo Magidin wrote:
> In article <S4prd.9967$d5.89607@newsb.telia.net>,
> Stefan Fredriksson <mcriley@telia.com> wrote:
>>
>>Let G be a subgroup of order 3 in S_4. Show that G can not be a normal
>>subgroup.

Thanks for your reply Arturo,

> How much do you know about S_4?

In the example? Nothing really.

> If you want to be very explicit: a subgroup of order 3 must be cyclic,

Because 3 is a prime, right?

> generated by a 3-cycle, (a,b,c). Let d be the element of {1,2,3,4}
> which is not in {a,b,c}. Then conjugate (a,b,c) by the permutation
> (a,d).

Sorry but my english is not very good. What do i do when I "conjugate"
(a,b,c) by the permutation (a,d) and why?

> This gives you (d,b,c), which is not in the subgroup generated
> by (a,b,c). So <(a,b,c)> cannot be normal. 

-- 
Regards
Stefan

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