Understanding Span: Multiple Elements in <x>

[kjartan]

If we call <a> the "span" of a, then I need some clarification on the concept of span.

def. if G is a group and a∈G, then <a> denotes the set of all integral powers of a. Thus,
<a> = {a^n : n∈ℤ}

thm. Let S be any subset of a group G, and let <S> denote the intersection of all of the subgroups of G that contain S. Then <S> is the unique smallest subgroup of G that contains S, in the sense that:
(a) <S> contains S
(b) <S> is a subgroup
(c) if H is any subgroup of G that contains S, then H contains <S>


Given these as a basis for interpreting <x>, how am I to read something like <[12], [20]>, for example, in ℤ_40? (where [k] is the congruence class to which k belongs, mod n).
I don't think I understand how to interpret the fact that more than one element is in the "span." How would I list out the elements in the set equal to this span?


Another example<p_H, p_V> with respect to the group of symmetries of a square (where p_H denotes a horizontal flip, and p_V a flip about the vertical axis). If I read this in light of the thm. about <S>, then I don't really know how to interpret what set of elements the span is equal to.

Could someone please help me to clear this up? Thanks!

 

[kjartan]

Well, I think I see how the first example should be read.

<[12],[20]> = <[4]> in ℤ_40, since

<[12],[20]> ⊆ <[4]> since [12] = [4]⊙[3] and [20] = [4]⊙[5]

<[12],[20]> ⊇ <[4]> since [4] = [12]⊙[2] ⊖ [20]⊙[1]

Hopefully my thinking is correct here. Then, given <[x],...,[y]>, we find the gcd of the elements in the span to get our spanning set.


On the other hand, I'm still not too sure about how to look at the subgroup <p_H, p_V> with respect to the group of symmetries of a square.