There is a line in my text 

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> This shows that all such products of integral powers of $a$ and $b$ form a subgroup of $G$...
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What does 'integral powers' mean here?

Additionally, there is an example that says $\{2,3\}$ generates $\mathbb Z\_6$ since 

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> $2+3 = 5$, so that any subgroup containing $2$ and $3$ must contain $5$
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I understand that if the set contains $5$ it has to generate $\mathbb Z\_6$, but why do you take the step $2+3$?

The notation $\{2,3\}$ suggests that you take powers of $2$ under $+\_6$ and *separately* take powers of $3$ under $+\_6$.

In other words, is it ok to do the following?
$\{2,3,4\}$ generates $\mathbb Z\_6$ since

1. $2+\_63 = 5$
2. $3+\_64 = 1$
3. $2+\_64 = 0$

and hence the set $\{2,3,4\}$ generates $\mathbb Z\_6$? 
Also, is it incorrect to say that 'the product of the integral powers of the elements of the set $\{2,3,4\}$ generates $\mathbb Z\_6$?