A question about the search for normal subgroups- Conjugacy classes ?

[nalkapo]

A question about the search for normal subgroups- Conjugacy classes!?

Homework Statement

Aut(Z*_24) has Conjugacy classes of order 1, 21, 24, 24, 42 and 56.
Show that Aut(Z*_24) is simple.

Note: Aut(Z*_24) is sometimes written as U(Z_24)
Thanks for any idea or answer...

Homework Equations

The Attempt at a Solution

I have no idea! At first, why 24 is written 2 times?

 

[clamtrox]

It means that it has six conjugacy classes, two of which have the order 24. For starters, what's the definition of a simple group? And how do conjugacy classes relate to that definition?

 

[nalkapo]

Simple group means, if it contains no non-trivial normal subgroups. so, for example, if a group has prime order, then it has no nontrivial normal subgroup.
and conjugacy classes means for a and b in G, there exist an element g for which,
g.a.g^(-1)=b
that means g.a=g.b
also i know that, Aut(Z*_24) has 168 elements and this is the total of this conjugacy classes. for next step, what should I do?
Actually i want to know, how can I find conjugacy classes? how did they find this number of conjugacy classes?

 

[clamtrox]

So is there perhaps a connection between the normal subgroups and conjugacy classes? In particular, what would it mean in terms of conjugacy classes, if you had a nontrivial normal subgroup?

 

[nalkapo]

I don't understand what is conjugacy classes and how to compute them.. only i know its definition.