The Mathematical Symbol "Contains As Normal Subgroup or Equal To (⊵)"

The ⊵ Symbol in Mathematics: Contains As Normal Subgroup or Equal To

In the intricate tapestry of mathematical symbols, ⊵, or "Contains As Normal Subgroup or Equal To", occupies a special place, especially in the realm of group theory. This article demystifies its meaning and usage with the aid of a couple of examples.

Usage

The ⊵ symbol is typically employed in group theory to represent that a group either contains another group as a normal subgroup or is equal to it.

Examples

  • Example 1: In terms of group theory:
    Let \( G \) be a group and \( H \) and \( K \) be subgroups of \( G \). If \( H \) is contained as a normal subgroup within \( K \) or is equivalent to \( K \), it's denoted as \( H ⊵ K \).
  • Example 2: Using another subgroup scenario:
    Given a group \( G \) and its subgroups \( A \) and \( B \). If \( A \) is either a normal subgroup of \( B \) or is the same as \( B \), we symbolize this relation as \( A ⊵ B \).

To wrap up, the ⊵ symbol provides an efficient means to convey the relationship of one group being a normal subgroup of another or being equal to it. Familiarity with this notation is crucial for those delving deep into group theory.

Mathematical symbol 'Contains As Normal Subgroup or Equal To'

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Codes for the ⊵ Symbol

The Symbol⊵
Alt CodeAlt 8885
HTML Code⊵
HTML Entity⊵
CSS Code\22B5
Hex Code⊵
UnicodeU+22B5

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 8885. 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: &#8885;</b>My symbol: ⊵

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

HTML TextOutput
<b>My symbol: &rtrie;</b>My symbol: ⊵

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

CSS and HTML TextOutput
<style>
span:after {
content: "\22B5";}
</style>
<span>My symbol:</span> 		My symbol: ⊵

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

HTML TextOutput
<b>My symbol: &#x22B5;</b>My symbol: ⊵
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22B5 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+22B5. 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
22B5
[Hold down Alt]
[Press x]		⊵
(The 22B5 turns into ⊵. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22B5. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22B5"
document.write("My symbol: " + str)		My symbol: ⊵