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Projective Special Unitary Group

The projective special unitary group PSU_n(q) is the quotient group of the special unitary group SU_n(q) by its group center,

PSU_n(q)=SU_n(q)/{aI:a in F_(q^2)^×, a^(q+1)=a^n=1}.
(1)

Here I is the n×n identity matrix. The group order of the group center is d=gcd(n,q+1), so the group order of PSU_n(q) is

|PSU_n(q)|=1/dq^(n(n-1)/2)product_(j=2)^n(q^j-(-1)^j).
(2)

Here |G| denotes the group order of a finite group G. For n>=2, the group PSU_n(q) is a non-Abelian group and is simple, except for

PSU_2(2)=S_3
(3)
PSU_2(3)=A_4
(4)
PSU_3(2)=3^2:Q_8.
(5)

In the notation of the Atlas of Finite Groups, the cases that are simple are denoted U_n(q). Also, PSU_2(q)=PSL_2(q).

See also

General Unitary Group, Projective General Unitary Group, Projective Special Linear Group, Projective Special Orthogonal Group, Special Unitary Group

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References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GU_n(q), SU_n(q), PGU_n(q), and PSU_n(q)=U_n(q)." §2.2 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. x, 1985.Grove, L. C. Classical Groups and Geometric Algebra. Providence, RI: American Mathematical Society, 2002.

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Projective Special Unitary Group

Cite this as:

Weisstein, Eric W. "Projective Special Unitary Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProjectiveSpecialUnitaryGroup.html

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