The Mathematical Symbol "Normal Subgroup of or Equal To (⊴)"

The "Normal Subgroup of or Equal To" Symbol (⊴)

In the realm of group theory, a branch of abstract algebra, the symbol ⊴ is used to indicate that one group is either a normal subgroup of another or is equal to it.

Decoding ⊴

A subgroup \( H \) of a group \( G \) is said to be normal if, for every element \( a \) in \( G \) and every element \( h \) in \( H \), \( a * h * a^{-1} \) is also in \( H \). The ⊴ symbol is used to express that a group is normal or equivalent to another group.

Example 1: Simple Groups

Suppose \( G \) is a group and \( H \) is its subgroup. If \( H \) is normal in \( G \) or \( H \) is equal to \( G \), then we write: \[ H ⊴ G \]

Example 2: Illustrating Equivalence

If \( H \) is indeed equivalent to \( G \), the relation can be represented as: \[ H ⊴ G \] But also: \[ H = G \]

Applications of ⊴

The ⊴ symbol is pivotal in various areas, such as:

  • Mathematics: Particularly in abstract algebra and group theory, to express normal subgroup relationships.
  • Research: In academic papers and texts that involve group theoretical concepts.

In essence, the ⊴ symbol plays a fundamental role in conveying relationships between groups in a mathematical context, ensuring that the properties and structure of groups can be accurately described.

Mathematical symbol 'Normal Subgroup of or Equal To'

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Codes for the ⊴ Symbol

The Symbol⊴
Alt CodeAlt 8884
HTML Code⊴
HTML Entity⊴
CSS Code\22B4
Hex Code⊴
UnicodeU+22B4

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 8884. 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: &#8884;</b>My symbol: ⊴

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

HTML TextOutput
<b>My symbol: &ltrie;</b>My symbol: ⊴

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

CSS and HTML TextOutput
<style>
span:after {
content: "\22B4";}
</style>
<span>My symbol:</span> 		My symbol: ⊴

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

HTML TextOutput
<b>My symbol: &#x22B4;</b>My symbol: ⊴
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22B4 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+22B4. 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
22B4
[Hold down Alt]
[Press x]		⊴
(The 22B4 turns into ⊴. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22B4. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22B4"
document.write("My symbol: " + str)		My symbol: ⊴