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# Group theory question watch

1. I need to show if G is a finite group and x is in G then x^-1 is in {1, x, x^2, ...}

Is this line of reasoning OK:

since G is finite, so must {1, x, x^2, ...}

so let t be the order of x:

{1, x, x^2, ... x^(t-1)}

Since x*x^(t-1) = 1, then x^(t-1) = x^-1
2. Exactly the right reasoning but you can say it slightly better:

Since G is finite and contains x, the cyclic group generated by x, <x> is a subgroup and is finite.

<x> is therefore a group containing x, and so must also contain x^-1 by group axioms.
3. (Original post by JoMo1)
Exactly the right reasoning but you can say it slightly better:

Since G is finite and contains x, the cyclic group generated by x, <x> is a subgroup and is finite.

<x> is therefore a group containing x, and so must also contain x^-1 by group axioms.
thanks that makes lots of sense

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Updated: March 31, 2011
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