Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general âŚ
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of nonâabelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and DavisâXiang family and their amorphic abelian Cayley schemes to produce amorphic nonâabelian Cayley schemes.
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results âŚ
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained âŚ
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords âŚ
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference âŚ
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ âŚ
Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions $xy^{-1}$ for all distinct two elements $x,y\in D$, represent each non-identity element in $G$ exactly $\lambda$ times. A $(v,k,\lambda)$-signed difference set is a generalization of a $(v,k,\lambda)$-difference set $D$, which satisfies all properties of $D$, but has a sign for each element in $D$. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes.
Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ âŚ
Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions $xy^{-1}$ for all distinct two elements $x,y\in D$, represent each non-identity element in $G$ exactly $\lambda$ times. A $(v,k,\lambda)$-signed difference set is a generalization of a $(v,k,\lambda)$-difference set $D$, which satisfies all properties of $D$, but has a sign for each element in $D$. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference âŚ
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained âŚ
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups.
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords âŚ
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general âŚ
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of nonâabelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and DavisâXiang family and their amorphic abelian Cayley schemes to produce amorphic nonâabelian Cayley schemes.
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results âŚ
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type âŚ
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups Z 4 2 k Ă Z 4 4 l â 4 k for all k when â is odd and for all k < â when â is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all k when â is even and for all k<â when â is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, θ) and C p (m, θ, Ͼ).
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, θ) and C p (m, θ, Ͼ).
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small âŚ
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).
We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are âŚ
We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and Ď any character of G, Ď(D) -ÎŁĎ>x(ÎŻ7) âŚ
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and Ď any character of G, Ď(D) -ÎŁĎ>x(ÎŻ7) is an algebraic integer of absolute value V~n in the field of mth roots of 1, where m is the order of Ď.Known facts about such integers and the relations which the Ď(D) must satisfy (as Ď varies) may yield information about D by the Fourier inversion formula.In particular, if Ď(D) is necessarily divisible by a relatively large integer, the number of elements g of D for which Ď(g) takes on any given value must be large; this yields some nonexistence theorems.Another theorem, which does not depend on a magnitude argument, states that if n and v are both even and Îą, the power of 2 in v, is at least half of that in n, then G cannot have a character of order 2 Îą , and thus G cannot be cyclic.A difference set with v = An gives rise to an Îadamard matrix; it has been conjectured that no such cyclic sets exist with v > 4.This is proved for n even by the above theorem, and is proved for various odd n by the theorems which depend on magnitude arguments.In the last section, two classes of abelian, but not cyclic, difference sets with v = in are exhibited.A subset D of a finite group G is called a difference set if every element ÎŚe of G can be represented in precisely Îť ways as d x d^, d { e D. If Ď is any nonprincipal character of G, we must then have | ^jdeD Ď(d) \ -Î/ÎÎŞ, n -k -Îť, where k is the order of D. We shall write Ď(D) for ÎŁidβD lid) (as in [8]).If G is abelian and | Ď{D) \ = V~n for some subset D and all nonprincipal characters of G, D is a difference set in G.This work originated in a search for difference sets with G cyclic of order v, and the parameters related by v = in.Because in this case every divisor of n is a divisor of v, Hall's theorem on multipliers, [5], one of the main tools in the study of difference sets, cannot be applied.The method presented here is particularly suitable for computation of difference sets if v and n have common factors.It is roughly as follows: the numbers X(D) are algebraic integers of absolute value V~n in the field of mth roots of 1, where m is the order of Ď (as an
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field âŚ
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field of order pn, GF(pn).The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (xi, Xi, x3) and (kxh kx2, kx3) are to be thought of as the same point.The totality of points whose coordinates satisfy the equation uiXi+u2x2+u3x3 = 0, where ui, w2, u3 are marks of the GF(pn), not all zero, is called a line.The plane then consists of p2n+pn+l = q points and q lines; each line contains pn + \ points, j A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer Âť, and there is only one such PG(2, pn) for a given p and Âť (VB, p. 247, VY, p. 151).Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane.(A collineation is a 1-1 transformation carrying points into points and lines into lines.)Suppose C carries A o into A\, Ax into * Presented to the Society, October 27, 1934, under a different title;
Abstract A partial difference set (PDS) having parameters ( n 2 , r ( n â1), n + r 2 â3 r, r 2 â r ) is called a âŚ
Abstract A partial difference set (PDS) having parameters ( n 2 , r ( n â1), n + r 2 â3 r, r 2 â r ) is called a Latin square type PDS, while a PDS having parameters ( n 2 , r ( n +1), â n + r 2 +3 r, r 2 + r ) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2âgroups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. Š 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266â282, 2009
Abstract Latin square type partial difference sets (PDS) are known to exist in R Ă R for various abelian p âgroups R and in ⤠t . We construct a âŚ
Abstract Latin square type partial difference sets (PDS) are known to exist in R Ă R for various abelian p âgroups R and in ⤠t . We construct a family of Latin square type PDS in ⤠t à ⤠2 nt p using finite commutative chain rings. When t is odd, the ambient group of the PDS is not covered by any previous construction. Š 2002 Wiley Periodicals, Inc. J Combin Designs 10: 394â402, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10029
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the âŚ
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of DingâYuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked âŚ
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2âgroups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in nonâ2âgroups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2âgroups. We give a new construction for linking systems of difference sets in 2âgroups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2âgroups, new linking systems in other 2âgroups for which no system was previously known, and the first known examples in nonabelian groups.
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general âŚ
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the DavisâXiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of nonâabelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and DavisâXiang family and their amorphic abelian Cayley schemes to produce amorphic nonâabelian Cayley schemes.
An [n, k, n - k + 1] linear code is called an MDS code. An [n, k, n - k] linear code is said to be almost maximum distance âŚ
An [n, k, n - k + 1] linear code is called an MDS code. An [n, k, n - k] linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system S(5, 6, 12) with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for t ⼠2 remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over GF(3 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> ) holding an infinite family of 3-designs and an infinite family of near MDS codes over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2s</sup> ) holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal.
<p style='text-indent:20px;'>Combinatorial <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code âŚ
<p style='text-indent:20px;'>Combinatorial <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>-design. Till now only a small amount of work on constructing <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-designs from codes has been done. In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters satisfying the generalized Kasami case, respectively. We then obtain infinite families of <inline-formula><tex-math id="M4">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters. In particular, the codes derived from the dual of binary BCH codes hold five <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-designs when <inline-formula><tex-math id="M6">\begin{document}$ m = 4 $\end{document}</tex-math></inline-formula>.</p>
The concept of directed strongly regular graphs was introduced by Duval in \A Directed Graph Version of Strongly Regular Graphs [Journal of Combinatorial Theory, Series A 47 (1988) 71 { âŚ
The concept of directed strongly regular graphs was introduced by Duval in \A Directed Graph Version of Strongly Regular Graphs [Journal of Combinatorial Theory, Series A 47 (1988) 71 { 100]. Duval also provided several construction methods for directed strongly regular graphs. We construct several new classes of directed strongly regular graphs with parameters = = t 1 or +1 = = t. The directed strongly regular graphs reported in this paper are obtained using a block construction of adjacency matrices of regular tournaments and circulant matrices. We then give some algebraic and combinatorial interpretation of these graphs in connection with known directed strongly regular graphs and related combinatorial structures.
The question as to whether there exists an infinite family of near MDS codes holding an infinite family of t-designs for t ⼠2 was answered in the recent paper âŚ
The question as to whether there exists an infinite family of near MDS codes holding an infinite family of t-designs for t ⼠2 was answered in the recent paper [Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory 66(9) (2020)], where an infinite family of near MDS codes holding an infinite family of 3-designs and an infinite family of near MDS codes holding an infinite family of 2-designs were presented, but no infinite family of linear codes holding an infinite family of 4-designs was presented. Hence, the question as to whether there is an infinite family of linear codes holding an infinite family of 4-designs remains open for 71 years. This paper settles this longstanding problem by presenting an infinite family of BCH codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+1</sup> +1 over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> ) holding an infinite family of 4-(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> + 1, 6, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> - 4) designs. This paper also provides another solution to the first question, as some of the BCH codes presented in this paper are also near MDS. Moreover, an infinite family of linear codes holding the spherical geometry design S(3, 5, 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + 1) is presented. The new direction of searching for t-designs with elementary symmetric polynomials will be further advanced.
A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called âŚ
A partial difference set having parameters $(n^2, r(n-1), n+r^2-3r,r^2-r)$ is called a Latin square type partial difference set, while a partial difference set having parameters $(n^2, r(n+1), -n+r^2+3r,r^2+r)$ is called a negative Latin square type partial difference set. In this paper, we generalize well-known negative Latin square type partial difference sets derived from the theory of cyclotomy. We use the partial difference sets in elementary abelian groups to generate analogous partial difference sets in nonelementary abelian groups of the form $(Z_p)^{4s} \times (Z_{p^s})^4$. It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number. We also give a generalization of subsets of Type Q, partial difference sets consisting of one fourth of the nonidentity elements from the group, to nonelementary abelian groups. Finally, we give a similar product construction of negative Latin square type partial difference sets in the additive groups of $(F_q)^{4t+2}$ for an integer $t \geq 1$. This construction results in some new parameters of strongly regular graphs.
We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters \[(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})\] for integers $q$ and $l$ ($q, l\ge 2$), âŚ
We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters \[(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})\] for integers $q$ and $l$ ($q, l\ge 2$), and \[(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)\] for all prime powers $q$ and $l\in \{1, 2,..., q\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4, 7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8, 14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on "Parameters of directed strongly regular graphs," at ${http://homepages.cwi.nl/^\sim aeb/math/dsrg/dsrg.html}$ by S. Hobart and A. E. Brouwer. We review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs.
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, âŚ
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, except that we do not exclude the possibility that two or more of the groups that we list may be isomorphic.The investigation is perhaps not without interest simply as an example of the use of linear methods in p-group theory; but the main motivation for it is that some result along these lines is needed by Suzuki in his classification [4] of ZT-groups.It is a pleasure to acknowledge that this paper is, in a direct way, a fruit of the special year in Group Theory organized by the Department of Mathematics at the University of Chicago.A 2-group with only one involution, that is, a eyelie or generalised quaternion group obviously has the property under discussion; and an abelian group has it if and only if it is a direct product of eyelie 2-groups all of the same order.It is convenient to exclude these eases from the beginning, and define a Suzulci 2-group as a non-abelian 2-group with more than one involution, having a eyelie group of automorphisms which permutes its involutions transi- tively.Evidently, the involutions of a Suzuki 2-group G all belong to its center, and so constitute, with the identity, an elementary abelian subgroup fh(G) of order q 2", n > 1.We shall show that fI(G) Z(G) q(G) G', so that G is of exponent 4 and class 2. The automorphism ( which permutes cyclically the q 1 involutions evidently has order divisible by q 1.We shall show that can be taken to have order precisely q 1, and so to be regular.The order of G is either q or qa.In many ways, it would be more satisfactory to impose on G the simpler, weaker condition that the involutions of G are permuted transitively by the full automorphism group of G. Possibly such a relaxation would not bring in any large class of new groups; but the condition seems to be very hard to handle.However, a little of our argument extends to the general ease, and this part has been stated for that ease.The methods used are similar to those involving the associated Lie ring (el.e.g. [2]),but we shall not construct this ring explicitly.The setup, which we shall presuppose, is as follows.If H is a subgroup of the 2-group G, and K a normal subgroup of H with elementary abelian factor group H/K,
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.