What exactly does $\langle a,b \rangle$ mean? Where a and b are elements of a group G?

Question

Does it mean either a or b will generate the whole of $\langle a,b \rangle$ or does it mean that some of the elements will be generated by a, some by b, and some by $a^rb^q$? The book I'm reading doesn't really make it very clear.

Answer 1

$H:=\langle a,b\rangle$ is formed by finite sequences of $a$ and $b$ i.e. $H = \lbrace e, a, b, aa, ab, ba, bb, aaa\ldots \rbrace$

(if $H$ is infinite you might also have to add $a^{-1}$ and $b^{-1}$ to the mix)

If $H$ is abelian, you can simply say that $H = \lbrace a^rb^s\ : r,s\in\mathbb{Z}\rbrace$.

Answer 2

$\langle a,b\rangle$ denotes the group genereated by $a$ and $b$ and contains any "word" made with the alphabet $a, a^{-1}, b$ and $b^{-1}$: a typical element can be written as a product $$a^{{e_1}}b^{{f_1}}a^{{e_2}}b^{{f_2}} \cdots a^{{e_k}}b^{{f_k}},$$where all the $e's$ and $f's$ are integers. So $b^{15}a^2b^{-14}$ would be an example, as well as $abab^2ab^{-1}$, etc. etc.