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unlhd

Represents a normal subgroup relation in abstract algebra and group theory, indicating a special hierarchical relationship between mathematical structures.

Overview

Commonly used in advanced mathematics to denote normal subgroup relationships, particularly in group theory and abstract algebra where hierarchical structures need to be precisely expressed.

  • Essential for describing group structure theorems and normal series
  • Often appears alongside \trianglelefteq as an alternative notation
  • Frequently used in research papers and advanced algebra textbooks
  • Particularly useful when discussing quotient groups and homomorphisms

Examples

Expressing a partial order relation in lattice theory.

A⊴B  ⟹  f(A)⊴f(B)A \unlhd B \implies f(A) \unlhd f(B) 
A \unlhd B \implies f(A) \unlhd f(B)

Showing subset relationships with additional structure in algebra.

H⊴G denotes that H is a normal subgroup of GH \unlhd G \text{ denotes that } H \text{ is a normal subgroup of } G 
H \unlhd G \text{ denotes that } H \text{ is a normal subgroup of } G

Comparing elements in a partially ordered set.

x⊴y⊴z  ⟹  x⊴zx \unlhd y \unlhd z \implies x \unlhd z 
x \unlhd y \unlhd z \implies x \unlhd z