If an element \(a\) of a group \(G\) has finite order \(n\), and \(a^m = e\) for some integer \(m\), then show that \(n | m\).

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If \(G\) is a finite abelian group and \(a, b \in G\), show that \(o(ab) | \text{lcm}(o(a), o(b))\).

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Let G be a finite group with more than one element. Show that G has an element of prime order . For Solution : Click here

Show that any finite group of even order must contain an element of order 2 . For Solution : Click here

Show that in a finite group of even order, the number of elements of order 2 is odd. For Solution : Click here

Suppose that G is a finite group with the property that every non identity element has prime order. If Z(G) is non trivial, prove that every non identity element of G has the same order . For Solution : Click here

If G is a finite group, then order of any element of G divides order of G . For Solution : Click here