Non-Identity Elements of Prime Order
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Suppose G is a finite group with the property that every non-identity element has prime order (D3 and D5 are examples of groups with this property). Show that if the center of G, Z(G), is not trivial, then every nonidentity element of G has the same order.
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Solution Summary
Non-Identity Elements of Prime Order are investigated.
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Proof:
we select two arbitrary nonidentity elements x and y in G. Suppose the order of x is p and the order of y is q. From the condition, p and q are prime numbers. We have the following 3 cases.
Case 1: Both x and y are in Z(G), then xy=yx. Suppose the order of xy is r, then we have
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