group theory question

Need help on aiii

what does it mean by conjugate? or do I just power all of y by a?

Also can someone tell me if I got these right please.

ai) a b^-1 a^2 b a b a^-1

aii) b^-3 a^-1 b^-1 a

Holy moly. Is that uni stuff?

yep

ai) a b^-1 a^2 b a b a^-1

aii) b^-3 a^-1 b^-1 a

Need help on aiii

From a bit of digging I suspect you are looking for the element: $aya^{-1}$

You should have a definition of the symbology in your notes somewhere.

Holy moly. Is that uni stuff?

so aii is a b^-1 a^-1 b^-3 right?

and thanks for aiii, but I will ask my lecturer just in case.

For b, what does it mean by conjugate? I cannot for the life of me find the definition in my notes!

There's always Google.

Two elements x,y are conjugate if there exists an element, say a, such that $ax=ya$ or putting it another way $axa^{-1}=y$, and I suspect with the notation used in your photo, that's $y=x^a$

so is it like aza^-1=z' = a^2 b^2 ab^-1 ab^-1 ?

Also for 3c, is it like we find aza^-1=z' and a^-1 za=z' and bzb^-1=z' and b^-1 zb=z'
??

Well, at the end z has a^-1 already, so you'd end up with a^-2 at the end

$aza^{-1}=z' = a^2 b^2 ab^{-1} ab^{-1}a^{-2}$



I suspect so. Those are certainly conjugate, but:

Conjugacy is an equivalence relation, and you could do $(ab)z(ab)^{-1}$ etc, but then there would be an infinite number of possibilities.

But since you're expected to list them all, that can't be the case.
I don't know what's expected there - I'd need to see some of the background material, or someone else may be familiar with this and more able to help.