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## Main Question or Discussion Point

a group or a cyclic group of finite order can i just repeatedly write down the repeated elements and form a very large even infinite group?

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a group or a cyclic group of finite order can i just repeatedly write down the repeated elements and form a very large even infinite group?

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HallsofIvy

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why" If the group is of finite order, then every member of it has finite order."? The group may not be a cyclic group. For example {1,2,3} the group is finite the member may have infinte order like the element {2}?

"get you back to the identity and then you get nothing new." is what i am confused; i am always wondering if i can write a group called {1,1,1,1,1,1,1,1,1,1,1,1.................}. Please help. Thanks

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HallsofIvy

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In any case, the proof that every element of a finite group has finite order is elementary. I would be surprised if you didn't see it shortly after learning the definition of "order".

Let a be any member of the finite group, G. Then, for all n, [itex]a^n\in G[/itex]. Since there are only a finite number of members of G while there are an infinite number of positive integers, by the pigeon hole principal, we must have [itex]a^k= a^j[/itex] for some distinct j and k. If j< k, then [itex]a^ka^{-j}= a^{k- j}= a^ja^{-j}= e[/itex] where e is the group identity and I am using "[itex]a^{-j}[/itex] to represent the inverse of [itex]a^j[/itex]. If k< j, then [itex]a^ka^{-k}= e= a^{j}a^{-k}= a^{j-k}[/itex].

In either case, a has finite order.

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lavinia

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But if you just require that each element be of finite order then one can have an infinite group with no problem

take the infinite direct product of Z/2 with itself.

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