# Physics Notes by Jakob Schwichtenberg

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group_theory:notions:subgroups

# Subgroups

A subgroup $H$ of a given group $G$ consists of elements of $G$ that have some additional property.

For example, the subgroup $SO(N)$ of $O(N)$ consists of all $N \times N$ matrices with determinant equal to $1$. ($O(N)$ consists of all $N \times N$ matrices $M$ that fulfil the condition $M^T M = 1$. $SO(N)$ consists of all $N \times N$ matrices $M$ that fulfil the conditions $M^T M = 1$ and $\det(M) =1$.)

The mathematical notation to indicate that some group $H$ is a subgroup of another group $G$ is

$$H \subset G .$$

## Normal Subgroups

[A] normal subgroup [is] a subgroup that “looks the same from every perspective.” For example, the subgroup of translations in the Euclidean group is always normal because the description “$g$ is a translation” is the same from every perspective (that is, it's invariant under conjugation).