Definition:Normal Subgroup

Definition

Let $G$ be a group and $H \le G$.

Then the subgroup $H$ is called a normal subgroup of $G$ iff:

$\forall g \in G: g H = H g$

where $g H$ and $H g$ are the left and right cosets respectively of $g$ modulo $H$.

This is represented symbolically as $H \triangleleft G$.

Clearly, by mutiplying the above definition on either side by $g^{-1}$, this can be stated equivalently as:

$H \triangleleft G := \forall g \in G: g H g^{-1} = H = g^{-1} H g$

or, to use the notation introduced in the definition of the congugate:

$H \triangleleft G := H^g = H $

A general normal subgroup is usually represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).

Also known as

Some sources refer to a normal subgroup as an invariant subgroup or a self-conjugate subgroup.

Also see

• Normal Subgroup Equivalent Definitions
• Normal Subgroup Test

• Results about normal subgroups can be found here.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.6$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 11$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.10$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 46$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 49$
• John F. Humphreys: A Course in Group Theory (1996): $\S 7$: Definition $7.3$