A weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)∼φ(x)φ(y) for all x,y∈G. Here ∼ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a ...group (G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x↦x
−1
. Let
0
(G) = ⟨Aut(G),I⟩≤ (G) and say that G has trivial weak Cayley table group if (G) =
0
(G). We show that PSL(2,p
n
) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.
A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ∼ φ(x)φ(y) for all x, y ∈ G. Here ∼denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: ...G → G forms a group (G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x
−1
. Let
0
(G) = ⟨Aut(G), I⟩ ≤ (G) and say that G has trivial weak Cayley table group if (G) =
0
(G). We show that all finite irreducible Coxeter groups (except possibly E
8
) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.
We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible ...left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d
−2
where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS
3
), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.
The set of subspaces of a given dimension in an attenuated space has a structure of a symmetric association scheme and this association scheme is called an association scheme based on an attenuated ...space. Association schemes based on attenuated spaces are generalizations of Grassmann schemes and bilinear forms schemes, and also
q
-analogues of nonbinary Johnson schemes. Wang, Guo, and Li computed the intersection numbers of association schemes based on attenuated spaces. The aim of this paper is to compute character tables of association schemes based on attenuated spaces using the method of Tarnanen, Aaltonen, and Goethals. Moreover, we also prove that association schemes based on attenuated spaces include as a special case the
m
-flat association scheme, which is defined on the set of cosets of subspaces of a constant dimension in a vector space over a finite field.
Previously, the author made the following conjecture: if a finite group has two semiproportional irreducible characters
φ
and
ψ
, then
φ
(1) =
ψ
(1). In the present paper, a new confirmation of the ...conjecture is obtained. Namely, the conjecture is verified for symplectic groups Sp
4
(
q
) and PSp
4
(
q
).
The second largest sporadic simple group, the “Baby Monster,” has a maximal subgroup
2
+
1
+
22
.
Co
2
which is a non-split extension of an extraspecial group of order 2
23 by the second sporadic ...simple Conway group
Co
2
. The construction of the character table of this group is described.
The investigation of nonclassical interactions (orthogonality relations) in finite groups is continued. A description of all small interactions (small
D
-blocks and small Φ-blocks) in symplectic ...groups Sp
4
(
q
) for even
q
is obtained. In particular, two known conjectures on these interactions are confirmed for these groups. Note that the existence of a small
D
-block in a group is equivalent to the existence of a pair of semiproportional rows in its character table and the existence of a small Φ-block in a group is equivalent to the existence of a pair of semiproportional columns in its character table.
In this paper the problem of a particle in an array of hexagons with periodic boundary condition is solved. Using the projection operators, we categorize eigenfunctions corresponding to each of the ...irreducible representations of the symmetry group . Based on these results, the Dirichlet and Neumann boundary conditions are discussed.
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