About the number of their conjugacy classes in some classes of finite simple groups

Question

We know that the orders of simple groups $B_n(q)$ and $C_n(q)$ are equal. What about the number of their conjugacy classes? Are they equal or not?

Any reply, comment, remark or reference is appreciated in advance.

Answer 1

As Stefan says, there is a clear counterexample for the simple groups of these two types, which can equally well be confirmed by looking at the relevant character tables in the Atlas of Finite Groups (Oxford, 1985). It may be useful to add some further remarks and references, to indicate how general this discrepancy is.

1) The simple groups you call $B_n(q)$ and $C_n(q)$ are actually isomorphic if $q$ is a power of 2, even though the ambient algebraic groups are not isomorphic in characteristic 2: they are instead related by one of Chevalley's "special" isogenies over an algebraically closed field.

2) The matrix groups are (respectively) $\mathrm{SO}_{2n+1}$ and $\mathrm{PSp}_{2n}$, each of index 2 in a universal group which is (respectively) a spin group or a symplectic group. The fact that there are closely related nonsimple groups complicates the study of both classes and characters.

3) For these and other classical groups over fields, G.E. Wall worked out a concrete description of conjugacy classes along with an algorithmic method to determine the total number of classes here. Later Lusztig made a careful, but rather technical, study of the ordinary characters for such groups here (with many details about character values added in later papers). In his $\S6$ he discusses with references the number of conjugacy classes in finite classical groups; but care must be taken about precisely which groups he deals with.

4) It's true that your two finite groups have the same order for any odd $q$, even though they aren't isomorphic when $n \geq 3$. It's also true, by general theorems of Steinberg, that both groups have the same number of semisimple classes (classes consisting of elements of order not divisible by $p$) and even have the same total number of unipotent elements (elements of order a power of $p$). However, they don't have the same number of unipotent classes, as seen in various sources such as Carter's book Finite Groups of Lie Type (Wiley, 1985). For arbitrary $n$ there is no single closed expression for the total number of classes (say as a polynomial in $q$), but the papers by Wall and Lusztig give a good indication as to why the answer will differ for your two groups. But keep in mind that you have to require $q$ to be odd and $n \geq 3$.

Answer 2

No, in general the numbers of conjugacy classes of $B_n(q)$ and $C_n(q)$ differ -- for example, $B_3(3)$ has $58$ conjugacy classes, while $C_3(3)$ has $74$. This can be found easily with GAP:

gap> Size(CharacterTable("O7(3)"));
4585351680
gap> Size(CharacterTable("S6(3)"));
4585351680
gap> NrConjugacyClasses(CharacterTable("O7(3)"));
58
gap> NrConjugacyClasses(CharacterTable("S6(3)"));
74