I'm trying to find an example of a group $G$ such that $|G| = 120$, and a non-normal subgroup $H$ within it. Of course, my first instinct is to let $G = S_5$, but this doesn't work because all subgroups of $S_5$ are normal. Help would be greatly appreciated!
Example of a Group with Certain Characteristics
1
$\begingroup$
group-theory
-
7Oh, dear: not at all! The only non-trivial normal subgroup $\,S_5\,$ has is $\,A_5\,$...**all** its other non-trivial subgroups are non-normal – 2012-12-16
-
0If I take the all the permutations with cycle type 5, say {(12345), (12354),(12435),...}, why is this not normal? – 2012-12-16
-
0It's a cyclic Sylow sub group of A_5 of order 5, and all the above are conjugate so the conjugacy class is not of order 1, so not normal – 2012-12-16
-
2@CardFlower, conjugation classes are not *usually* subgroups! – 2012-12-16
-
0And oups. I meant <(12345)> obviously, and its 6 (i think) conjugacy classes. – 2012-12-16
-
0I think the question, @CardFlower, is *why is this normal?* :-) – 2012-12-16
-
0I thought that conjugacy by any element in $S_5$ preserves cycle type, so that regardless of what you hit $<(12345)>$ with, it will stay within $<(12345)>$. – 2012-12-16
-
0Conjugating rho=(12345) will preserve its cycle type, but the result will not necessarily be a power of (12345). (Do some computations for yourself to check.) As it is a 5-cycle, the subgroup generated by it will have 5 elements, whereas the conjugacy class associated to it has 5!/5=4!=24 members. As Don noted, conjugacy classes are not the same thing as subgroups, though conjugacy classes of the symmetric groups are always those permutations of a given cycle type as you observe. – 2012-12-16