Semidirect Product - Notation

Notation

Usually the semidirect product of a group H acting on a group N (in most cases by conjugation as subgroups of a common group) is denoted by or . However, some sources may use this symbol with the opposite meaning. In case the action should be made explicit, one also writes . One way of thinking about the symbol is as a combination of the symbol for normal subgroup and the symbol for the product .

Unicode lists four variants:

		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.