Discussion
Consider the following statements :
(I) Let G be a finite group. Then every element is of finite order.
(II) Let G be a group such that every element is of finite order. Then G is a finite group.
Which is true ?
Answer: A
Only (I) is true.
(I) In a finite group of order \(n\), the powers \(g, g^{2}, g^{3}, \ldots\) cannot all be distinct, so \(g^{i} = g^{j}\) for some \(i \lt j\) and \(g^{j - i} = e\). By Lagrange's theorem the order of \(g\) even divides \(n\).
(II) A group in which every element has finite order (a torsion group) need not be finite. The group \(\mathbf{Q}/\mathbf{Z}\) is infinite, yet each element \(p/q + \mathbf{Z}\) has order dividing \(q\). An infinite direct sum of copies of \(\mathbf{Z}_{2}\) is another example: every non-identity element has order 2.
So the options asserting (II) are wrong, and 'neither' ignores the elementary argument for (I).
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