In a group G, let elements x and y satisfy yx = xyz where z is in the center of

Question

In a group G, let elements x and y satisfy yx = xyz where z is in the center of the group. a) Work out a formula of the type (xy)n = x?y?z? for all n ? 0. For instance, taking into account that z is in the center, (xy)2 = xyxy = x(xyz)y = x2yzy = x2y2z. Calculate at least through n = 4 before guessing the formula to prove by induction. Otherwise you will likely guess the wrong formula. (Hint: You can write yx = xyz as x?1yx = yz). b) Derive a formula of the form (xy)n = x?y?z? which is valid for all integers, including n < 0.

Explanation / Answer

the question marks next to xyz are their exponents, it is what needs to be found (so those are supposed to be there) and n? 0 should say n greater than or equal to zero the hint should say (x ^ -1)(yx) =yz

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