Group relations

The table below summarises the inclusion relations between frequently used groups

stateDiagram
	M_n(F)
	GL(n,F)
	SL(n,F)
	U(n,F)
	SU(n,F)
	M_n(R)
	O(n)
	SO(n)
	M_n(F) --> GL(n,F) : invertible matrices
	GL(n,F) --> SL(n,F) : determinant equals 1
	U(n,F) --> SU(n,F) : determinant equals 1
	M_n(R) --> O(n) : preserves inner products
	O(n) --> SO(n) : determinant equals 1
	GL(n,F) --> U(n,F) : preserves inner products
	SL(n,F) --> SU(n,F) : preserves inner products
	U(n,F) --> O(n) : real matrices only
	SU(n,F) --> SO(n) : real matrices only

Where, up to LaTeX formatting, we have:

• $\mathbb{F}$ is a field
• $\mathbb{R}$ is the real line
• $M_n(\mathbb{F})$ is the set of $n\times n$ matrices over $\mathbb{F}$
• $M_n(\mathbb{R})$ is the set of $n\times n$ matrices over $\mathbb{R}$
• $GL(n,\mathbb{F})$ is the General Linear Group
• $SL(n,\mathbb{F})$ is the Special Linear Group
• $U(n,\mathbb{F})$ is the Unitary Group
• $SU(n,\mathbb{F})$ is the Special Unitary Group
• $O(n)$ is the Orthogonal Group
• $SO(n)$ is the Special Orthogonal Group