## The Mathematical Symbol "Does Not Contain As Normal Subgroup (⋫)"

The "Does Not Contain As Normal Subgroup" Symbol (⋫): A Group Theory Perspective

In the realm of abstract algebra, particularly group theory, symbols play a pivotal role in conveying intricate relationships and structures. One such symbol is the "Does Not Contain As Normal Subgroup" notation, represented as ⋫.

## Role in Group Theory

In group theory, a branch of abstract algebra, the concept of a "normal subgroup" is central. A normal subgroup is a subgroup that is invariant under conjugation. The ⋫ symbol is used to indicate that one group does not contain another group as a normal subgroup. In other words, the subgroup is not invariant under conjugation by members of the larger group.

Example 1:

Let $$G$$ be a group and $$H$$ be a subgroup of $$G$$. If $$H$$ is not a normal subgroup of $$G$$, this relationship can be denoted as:

$$H ⋫ G$$

Example 2:

Consider the symmetric group $$S_3$$, which consists of all permutations of three elements. The subgroup consisting of the identity and a single 2-cycle (like (12)) is not a normal subgroup of $$S_3$$. Symbolically, this can be expressed as:

$$\langle (12) \rangle ⋫ S_3$$

Through these examples, the relevance and utility of the "Does Not Contain As Normal Subgroup" notation in group theory become clear. As with many symbols in abstract algebra, understanding ⋫ requires a firm grasp of the underlying concepts and structures it represents.

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## Codes for the ⋫ Symbol

 The Symbol ⋫ Alt Code Alt 8939 HTML Code ⋫ HTML Entity ⋫ CSS Code \22EB Hex Code ⋫ Unicode U+22EB

## How To Insert the ⋫ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ⋫ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8939. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML TextOutput
<b>My symbol: &#8939;</b>My symbol: ⋫

(Method 4) Use the HTML Entity Code (for webpages).

HTML TextOutput
<b>My symbol: &nrtri;</b>My symbol: ⋫

(Method 5) Use the CSS Code (for webpages).

CSS and HTML TextOutput
<style>
span:after {
content: "\22EB";}
</style>
<span>My symbol:</span>
My symbol: ⋫

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML TextOutput
<b>My symbol: &#x22EB;</b>My symbol: ⋫
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22EB to place the ⋫ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);
Output

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ⋫ is U+22EB. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:
TypeOutput
22EB
[Hold down Alt]
[Press x]

(The 22EB turns into ⋫. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22EB. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22EB"
document.write("My symbol: " + str)
My symbol: ⋫