Non-Normal Subgroups in Simple Groups

[spamiam]

What are some simple groups that have non-normal subgroups? The only example I can think of is the alternating group for n > 4.

 

[mathwonk]

all non abelian simple groups have non normal subgroups. i.e. they have even order, so they have elements of order 2, hence subgroups of order 2, which are necessarily non normal.

 

[spamiam]

Thanks for your reply, mathwonk!

all non abelian simple groups have non normal subgroups. i.e. they have even order...

Wait, all non-abelian simple groups have even order? Or did you mean that as an example?

...so they have elements of order 2, hence subgroups of order 2, which are necessarily non normal.

Okay, the element of order 2 follows from Cauchy's theorem, but why are these subgroups "necessarily" non-normal? Is it just because we were already assuming the group was simple or is there a deeper reason?

I was kind of hoping for some specific examples of known groups. The only examples of simple groups with which I'm familiar are $\mathbb{Z}/p\mathbb{Z}$ and $A_n$ for n > 4. Are there any other well-known ones?

Thanks again!

 

[micromass]

Wait, all non-abelian simple groups have even order? Or did you mean that as an example?

Yes, a finite simple group is either $$\mathbb{Z}_p$$ or have even order. This is the contents of the celebrated Feit-Thompson theorem.

Okay, the element of order 2 follows from Cauchy's theorem, but why are these subgroups "necessarily" non-normal? Is it just because we were already assuming the group was simple or is there a deeper reason?

That is correct. The subgroup is non-normal, because we assumed that the group was simple.

I was kind of hoping for some specific examples of known groups. The only examples of simple groups with which I'm familiar are $\mathbb{Z}/p\mathbb{Z}$ and $A_n$ for n > 4. Are there any other well-known ones?

Thanks again!

To my knowledge, example of simple groups are kind of tricky. The article http://en.wikipedia.org/wiki/List_of_finite_simple_groups gives a list of simple groups. The most intriguing of these groups is the so-called monster group, which is very big. It contains a (non-normal of course) subgroup which is also huge and is called the baby monster.

 

[spamiam]

This is the contents of the celebrated Feit-Thompson theorem.

Celebrated, eh? Looks like I might have to go celebrate with a textbook.

To my knowledge, example of simple groups are kind of tricky. The article http://en.wikipedia.org/wiki/List_of_finite_simple_groups gives a list of simple groups.

You weren't kidding! I could understand the construction of about 3 families out of that list. Thanks for the link!