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Let g be in a group G. (i) Can g have larger order than G has? Explain. (ii) Can g have infinite order while G is finite?

Solution

Let o(G) is finite. As order of any element g in G must divide o(G) i.e o(g)=d | o(G), Therefore o(G)=d can not be greater then o(G). If o(G) is infinite every elemnt of g has either finite order or infinite, in any case not greater than o(G).

If o(G) is finite, let g be an element in G, chosen arbitrarily. Then o(g)=d must divide o(G). Therefore o(g) can not be infinite.