The Mathematical Symbol "Does Not Contain As Normal Subgroup or Equal (⋭)"

The "Does Not Contain As Normal Subgroup or Equal" Symbol (⋭): Deepening Our Understanding of Group Theory

In abstract algebra and group theory, symbols serve as shorthand to express intricate relationships between algebraic structures. Among these symbols, the "Does Not Contain As Normal Subgroup or Equal" notation, represented as ⋭, holds a distinct place. This article explores the applications and significance of this symbol, bolstered with examples for clarity.

Deciphering ⋭

In the realm of group theory, a "normal subgroup" is a subgroup invariant under conjugation. The symbol ⋭ communicates that a particular subgroup neither equals the larger group nor is contained as a normal subgroup within the larger group. Essentially, it's a way of expressing non-equivalence and non-normal containment in one symbol.

Example 1:

Let's consider a group \( G \) and a subgroup \( H \). If \( H \) is neither equal to \( G \) nor a normal subgroup of \( G \), the relationship between the two can be expressed as:

\( H ⋭ G \)

Example 2:

Assuming the symmetric group \( S_4 \), which represents all possible permutations of four distinct elements, and a subgroup \( K \) derived from \( S_4 \). If \( K \) is neither equivalent to \( S_4 \) nor considered a normal subgroup of \( S_4 \), the relationship can be symbolized as:

\( K ⋭ S_4 \)

The ⋭ symbol is an integral part of group theory notation, allowing mathematicians to convey intricate subgroup relationships in a concise manner. As with most mathematical symbols, context and a thorough understanding of the underlying principles are paramount to grasp the exact meaning and implications of ⋭.

Mathematical symbol 'Does Not Contain As Normal Subgroup or Equal'

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Codes for the ⋭ Symbol

The Symbol⋭
Alt CodeAlt 8941
HTML Code⋭
HTML Entity⋭
CSS Code\22ED
Hex Code⋭
UnicodeU+22ED

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 8941. 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: &#8941;</b>My symbol: ⋭

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

HTML TextOutput
<b>My symbol: &nrtrie;</b>My symbol: ⋭

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

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

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

HTML TextOutput
<b>My symbol: &#x22ED;</b>My symbol: ⋭
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22ED 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+22ED. 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
22ED
[Hold down Alt]
[Press x]		⋭
(The 22ED turns into ⋭. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22ED. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22ED"
document.write("My symbol: " + str)		My symbol: ⋭