How Do Group Generators Relate to the Order of a Group?

[logarithmic]

I have 2 questions:

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 $$r=\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\in D_4$$ , 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?

 

[Office_Shredder]

If G=<a,b> then G is going to be infinite. In general assuming that you have placed conditions on the generators, the order of a and the order of b both divide the order of G (assuming G is finite) but that's just Lagrange's theorem and has nothing to do with the fact that they are generators

You can have generators with relatively small orders lead to groups with relatively large orders. For example for S_n, it's generated by the transposition (1 2) and by the n cycle (1 2 ... n) which have orders 2 and n, but the order of S_n is n!. On the other hand you have D_n which is also generated by elements of order 2 and n, but the order of D_n is only 2n.

Groups that are presented via their generators are pretty tough to work with usually