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: