= = = - For any group K, recall that Z(K) denotes the centre of K. Define an infinite sequence of groups by K (0) = K, K

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For Any Group K Recall That Z K Denotes The Centre Of K Define An Infinite Sequence Of Groups By K 0 K K 1 (103.75 KiB) Viewed 25 times

= = = - For any group K, recall that Z(K) denotes the centre of K. Define an infinite sequence of groups by K (0) = K, K(1) = K (0)/Z(K),...,K (0) ( K (0) K(i-1)/Z(K(i-1)),.... (i) Prove by induction on n that |G(n) = 1, where G is a group of order pn as before. = = (ii) Determine the smallest value of i such that |G(0)| = 1 when G is each of the following groups: the cyclic group of order 8; the quaternion group Q8 of order 8. (iii) Give an example of a group K (which will not be a p-group) such that K(1) # 1 for all i. Fully justify your answer.