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Projective General Unitary Group

The projective general unitary group PGU_n(q) is the quotient group of the general unitary group GU_n(q) by its group center. The group center consists of the scalar matrices aI for which a^(q+1)=1, where I is the n×n identity matrix. Therefore,

PGU_n(q)=GU_n(q)/{aI:a in F_(q^2)^×, a^(q+1)=1}.

For a finite group G, |G| denotes its group order. In particular,

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

The quotient group has a faithful group action on the subset of projective space represented by nonzero vectors v satisfying H(v,v)=0, where H is the preserved Hermitian form. The projective image of the special unitary group is the projective special unitary group, which is a normal subgroup of PGU_n(q).

See also

General Unitary Group, Hermitian Form, Projective General Linear Group, Projective General Orthogonal Group, Projective 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 General Unitary Group

Cite this as:

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

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