# Understanding Span: Multiple Elements in <x>

• kjartan
In summary, the conversation discusses the concept of "span" in the context of groups and subgroups. The definition states that <a> refers to the set of all integral powers of a in a group G. The theorem states that <S> is the unique smallest subgroup of G that contains a given subset S. The conversation also includes examples of interpreting <x> in different contexts, such as in ℤ_40 and the group of symmetries of a square. The process of finding the elements in the set equal to the span is discussed, with the example of <[12], [20]> = <[4]> in ℤ_40. The concept of taking the gcd of elements in the span to get the spanning
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.

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.

## What is a span?

A span is an HTML tag used to group and style inline elements. It does not create a line break or any other visual change, but allows for targeted styling of specific elements within a larger block of text.

## How do I use a span?

To use a span, simply wrap the desired elements with the opening and closing span tags. You can then apply CSS styles to the span tag to target those specific elements.

## Can I use multiple spans in one element?

Yes, you can use multiple spans within one element. This can be useful for targeting different elements within a larger block of text, or for creating more specific styling for certain elements.

## What is the purpose of using multiple spans in one element?

The purpose of using multiple spans in one element is to target and style specific elements within a larger block of text. This allows for more precise and flexible styling options.

## Are there any limitations to using spans?

There are no specific limitations to using spans, but it is important to use them in moderation and only when necessary. Overusing spans can make the code more complex and difficult to maintain.

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