I have 2 questions:(adsbygoogle = window.adsbygoogle || []).push({});

1. Can anything be said about the order of a group from the order of its generators (or vice versa)? E.g. if a group G = <a,b>, is there any theorem that says the order of elements a, b is divisible by the order of G, or maybe, if G = <a,b>, then the order of G is the product of the order of a and b, or something else to relate the order of G, a, and b?

2. If D_4 is the dihedral group of order 8 and [tex] r=\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\in D_4[/tex] , is there any quick way to see that (D_4)/<r> is not cyclic, without computing the order of each of the 4 elements of (D_4)/<r> and showing none of them are of order 4?

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# Homework Help: Groups and generators

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