I am a math hobbyist working by myself. I am currently reading Beachy and Blair: "Abstract Algebra" on conjugacy in groups. I wish to ensure I have understood the notion of conjugacy classes. To do this I am considering the following group

$\displaystyle D_8$ = {1, r, $\displaystyle r^2$, $\displaystyle r^3$, s, sr, s$\displaystyle r^2$, s$\displaystyle r^3$}

and am looking at what would be required to determine the conjugacy class of r.

Beachy and Blair's definition (adapted for r and $\displaystyle D_8$) of the conjugacy class of r is

{g$\displaystyle \in$G| there exists an element a such that g = $\displaystyle {ar}a^{-1}$}

Now as far as I can see to determine the conjugacy class of r from first principles is quite lengthy - we are looking for g such that there exists an a such that g = $\displaystyle {ar}a^{-1}$ for some a

So start looking at possibiity g = 1

So we search through the elements of $\displaystyle D_8$ for an a such that 1 = $\displaystyle {ar}a^{-1}$

Try a = 1

1 = 1.r.$\displaystyle 1^{-1}$ ... No

Try a = r

a = r.r..$\displaystyle r^{-1}$ ... No

and so on to test all elements of $\displaystyle D_8$

Then go through the same process for g = r and so on through all the elements of $\displaystyle D_8$

Have I interpreted the definition correctly ... can someone affirm this or correct me please.

================================================== =

Another issue is that the text by Eie and Chang: A Course in Abstract Algebra on page 87 write:

"The equivalence class of a under conjugation is called the conjugacy class containing a in G and is given by

[a] = {$\displaystyle {ga}g^{-1}$$\displaystyle \in$G | g$\displaystyle \in$G}"

This does not seem to be the same definition as Beachy and Blair since the set of all g such that g = $\displaystyle {ax}a^{-1}$ does not seem to me to be the same as the set of all $\displaystyle {ax}a^{-1}$

I note also that in Nicholson's book: "Introduction to Abstract Algebra", on page 349 he defines the conjugacy class of a $\displaystyle \in$ G as follows:

class a = {x$\displaystyle \in$| x is conjugate to a} = {$\displaystyle {ga}g{-1}$ | g$\displaystyle \in$G}

The second part of this definition is similar to Eie and Changs and seems different to the first part of the definition???

Can someone please clarify this?

Peter