The Mathematical Symbol "Contains As Normal Subgroup (⊳)"

The ⊳ Symbol in Mathematics: Contains As Normal Subgroup

In the realm of group theory, a branch of abstract algebra, the ⊳ symbol, or "Contains As Normal Subgroup", stands out as an essential notation.


The ⊳ symbol is utilized to denote that a given subgroup is a normal subgroup of another group. In formal terms, if \( H \) is a subgroup of \( G \) and every left coset of \( H \) in \( G \) is also a right coset of \( H \) in \( G \), then \( H \) is a normal subgroup of \( G \), denoted as \( H ⊳ G \).


  • Example 1: Center of a group:
    The center of a group \( G \), denoted \( Z(G) \), consists of all elements that commute with every element of \( G \). The center is always a normal subgroup of \( G \):
    \( Z(G) ⊳ G \)
  • Example 2: In terms of quotient groups:
    If \( H ⊳ G \), then the set of left cosets \( G/H \) forms a group called the quotient group or factor group.

In summary, the ⊳ symbol is paramount in understanding the structure of groups and the relationships between subgroups. Recognizing when a subgroup is normal is crucial for various operations and results in group theory.

Mathematical symbol 'Contains As Normal Subgroup'

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Codes for the ⊳ Symbol

The Symbol
Alt CodeAlt 8883
HTML Code⊳
HTML Entity⊳
CSS Code\22B3
Hex Code⊳

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 8883. 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: &#8883;</b>My symbol: ⊳

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

HTML TextOutput
<b>My symbol: &vrtri;</b>My symbol: ⊳

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

CSS and HTML TextOutput
span:after {
content: "\22B3";}
<span>My symbol:</span>
My symbol: ⊳

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

HTML TextOutput
<b>My symbol: &#x22B3;</b>My symbol: ⊳
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22B3 to place the ⊳ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);

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

The Unicode for ⊳ is U+22B3. 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:
[Hold down Alt]
[Press x]

(The 22B3 turns into ⊳. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22B3. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22B3"
document.write("My symbol: " + str)
My symbol: ⊳