- #1

kjartan

- 15

- 1

**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!