Zvonimir Janko

Groups of prime power order

This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

Coset Enumeration in Groups and Constructions of Symmetric Designs

Publisher Summary This chapter discusses coset enumeration in groups and constructions of symmetric designs. It presents a group in terms of generators and relations so that each point or block stabilizer, can be expressed with the same generators. The Coxeter–Todd coset enumeration method with respect to the subgroup gives the number of cosets and also gives the permutation representation of group with respect to the (right) cosets of the subgroup. The corresponding programs have been made by Hrabe De Angelis for each stabilizer subgroup. To construct the design means only to put together all these permutation representations according to the orbit structure matrix. In JANKO–TRAN a symmetric design was constructed whose full automorphism group is discussed.

The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs

AbstractA symmetric 2-(324, 153, 72) design is constructed that admits a tactical decomposition into 18 point and block classes of size 18 such that every point is in either 0 or 9 blocks from a given block class, and every block contains either 0 or 9 points from a given point class. This design is self-dual and yields a symmetric Hadamard matrix of order 324 of Bush type, being the first known example of a symmetric Bush-type Hadamard matrix of order 4n2 for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, λ=μ=72 that admits a spread of cocliques of size 18. The Bush-type Hadamard matrix of order 324 leads to two new infinite classes of symmetric designs with parameters

Bush‐type Hadamard matrices and symmetric designs

v = 324(289^m + 289^{m - 1} + \cdot \cdot \cdot + 289 + 1),{\text{ }}k = 153(289)^m ,{\text{ }}\lambda \;{\text{ = }}\;{\text{72(289)}}^m ,

A Block Negacyclic Bush-Type Hadamard Matrix and Two Strongly Regular Graphs

and

Cyclic 2-(91, 6, 1) designs with multiplier automorphisms

v = 324(361^m + 361^{m - 1} + \cdot \cdot \cdot + 361 + 1),{\text{ }}k = 171(361)^m ,{\text{ }}\lambda \;{\text{ = }}\;90{\text{(361)}}^m ,