Number of conjugacy classes in symplectic group of fixed degree over a finite field is PORC function of field size
From Groupprops
Statement
Suppose is an even natural number, i.e.,
for some natural number
. Then, there exists a PORC function
of degree
such that, for any prime power
, the number of conjugacy classes in the symplectic group
(i.e., the symplectic group of degree
over the finite field of size
) is
.
A PORC function is a polynomial on residue classes -- it looks like different polynomial functions on different congruence classes modulo a particular number. In this case, we only need to consider congruence classes modulo to define the PORC function. In fact, for a field size of
, the polynomial depends only on the value
.
Particular cases
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Possibilities for ![]() |
Corresponding congruence classes mod ![]() ![]() |
Corresponding polynomials in PORC function of ![]() ![]() |
More information |
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2 | 1 | 2 | 1 2 |
0 1 |
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See element structure of special linear group of degree two over a finite field (note that ![]() |
4 | 2 | 2 | 1 2 |
0 1 |
![]() ![]() |
See element structure of symplectic group of degree four over a finite field |
Related facts
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- Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size