## The Mathematical Symbol "Wreath Product (≀)"

The ≀ Symbol in Mathematics: The Wreath Product

In the diverse landscape of mathematical symbols, the ≀ symbol designates the "Wreath Product", a specialized operation in the realm of group theory. The wreath product allows for the combination of two groups in a specific manner, yielding a new group that incorporates structures from both. This article aims to shed light on the primary applications of the ≀ symbol, underpinned by a couple of illustrative examples.

## Usage

In group theory, the wreath product is a way of combining two groups, often denoted as "acting" groups and "acted upon" groups. The resultant group integrates aspects of both initial groups. The ≀ symbol succinctly represents this operation, conveying the intricacies of the operation without getting bogged down in the details.

## Examples

• Example 1: Consider two cyclic groups $$C_2 = \{e, a\}$$ and $$C_3 = \{e, b, b^2\}$$. The wreath product of these groups, represented as $$C_2 ≀ C_3$$, results in a group of order $$2 \times 3^2 = 18$$.
• Example 2: When analyzing permutation groups, the wreath product can be used to combine them. For instance, $$S_2 ≀ S_3$$ represents a particular combination of the symmetric groups $$S_2$$ and $$S_3$$.

In summary, the ≀ symbol is pivotal in the field of group theory, enabling mathematicians to denote the wreath product operation efficiently. Through this operation, intricate group structures can be analyzed, deepening our understanding of group interactions and behaviors.

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## Codes for the ≀ Symbol

 The Symbol ≀ Alt Code Alt 8768 HTML Code ≀ HTML Entity ≀ CSS Code \2240 Hex Code ≀ Unicode U+2240

## 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 8768. 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: &#8768;</b>My symbol: ≀

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

HTML TextOutput
<b>My symbol: &wreath;</b>My symbol: ≀

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

CSS and HTML TextOutput
<style>
span:after {
content: "\2240";}
</style>
<span>My symbol:</span>
My symbol: ≀

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

HTML TextOutput
<b>My symbol: &#x2240;</b>My symbol: ≀
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2240 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+2240. 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
2240
[Hold down Alt]
[Press x]

(The 2240 turns into ≀. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2240. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2240"
document.write("My symbol: " + str)
My symbol: ≀