Editing Semidirect product (section)

== Notation ==

Usually the semidirect product of a group {{math|''H''}} acting on a group {{math|''N''}} (in most cases by conjugation as subgroups of a common group) is denoted by {{math|''N'' ⋊ ''H''}} or {{math|''H'' ⋉ ''N''}}. However, some sources<ref name="Vinberg(2003)">e.g., {{cite book|author=E. B. Vinberg|title=A Course in Algebra|location=Providence, RI|publisher=American Mathematical Society|page=389|isbn=0-8218-3413-4|date=2003}}</ref> may use this symbol with the opposite meaning. In case the action {{math|''φ'' : ''H'' → Aut(''N'')}} should be made explicit, one also writes {{math|''N'' ⋊{{sub|''φ''}} ''H''}}. One way of thinking about the {{math|''N'' ⋊ ''H''}} symbol is as a combination of the symbol for normal subgroup ({{math|◁}}) and the symbol for the product ({{math|×}}). [[Barry Simon]], in his book on group representation theory,<ref name="Simon1996">{{cite book|author=B. Simon|title=Representations of Finite and Compact Groups|location=Providence, RI|publisher=American Mathematical Society|page=6|isbn=0-8218-0453-7|date=1996}}</ref> employs the unusual notation <math>N\mathbin{\circledS_{\varphi}}H</math> for the semidirect product.

[[Unicode]] lists four variants:<ref>See [https://www.unicode.org/charts/#symbols unicode.org]</ref>

: {| class="wikitable"
!   !! Value  !! MathML !! Unicode description
|-
| ⋉ || U+22C9 || ltimes || LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
|-
| ⋊ || U+22CA || rtimes || RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
|-
| ⋋ || U+22CB || lthree || LEFT SEMIDIRECT PRODUCT
|-
| ⋌ || U+22CC || rthree || RIGHT SEMIDIRECT PRODUCT
|}

Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.

In [[LaTeX]], the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌.