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I need to show closure under group operation and inverse.

I can do the inverse which is usually the hardest part, but I'm stuck on the grp op.

So let a in K and b in K, both have finite distinct conjugates. Their conjugates are in the group too. WTS is that ab in K too.

if a, b in K then xax

^{-1}= c in K and yay

^{-1}

consider xy(ab)(xy)

^{-1}note since xax

^{-1}and yay

^{-1}are finite and distinct then x and y are finite and distinct hence xy and its inverse is finite

hence xy(ab)(xy)

^{-1}in K

what do you think?