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Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.12.2018)
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Last update: prof. RNDr. Aleš Drápal, CSc., DSc. (01.06.2022)
The exam is both oral and written. The student is given one of the six exam areas to prepare in the written form. Consequently the student may be asked to give an additional explanation orally. |
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Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (18.12.2018)
V. D. Belousov: Osnovy teorii kvazigrupp i lup, Nauka, Moskva, 1967
H. O. Pflugfelder: Quasigroups and Loops: An Introduction, Heldermann Verlag, 1991
D. Keedwell and József Dénes: Latin Squares and their Applications, 2nd Edition, North Holland, 2015
M. Hall, Jr.: Theory of Groups, MacMillan Co., 1959
D. R. Hughes a F. C. Piper: Projective planes, Springer Verlag, 1973 |
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Last update: prof. RNDr. Aleš Drápal, CSc., DSc. (01.06.2022)
1. Free quasigroups. Reduced words and rewriting rules. Congruences of quasigroups and loops. Connections with blocks of the multiplication group. The group of inner mappings and its meaning with respect to the normality of a subloop. Standard generators.
2. Semisymmetry, MTS, STS, and their interpretation via parastrophy. Projective and affine STS. Prolongation and STS loops. HTS and its characterization via distributivity and commutativity of Moufang loops. Structure of finite commutative Moufang loops.
3. Nuclei of loops. Their characterization by autotopisms. A proof that nuclei are subloops. Center of a loop. The connection of the center with the inner mapping group. The center of the multiplication group. Its relationship to the normalizer of the inner mapping group. The normality of a nucleus as a consequence of the normality of the left or the right multiplication group. Coincidence and normality of the nuclei in Bol and Moufang loops.
4. Definition of affine and projective planes, k-nets and transversal designs. Transitions between affine and planar plane. Definitions of affine planes by means of additive group and a multiplicative loop. Definition of a quasifield and collineations. Semifields and nearfields. Dickson nearfields. Connections of finite nearfields and sharply 2-transitive permutation groups.
5. Deriving Bol identities by means of left and right inverse properties. The descriptions by twists of translations. Derivation and equivalence of Moufang identities. Extra loops and their characterization as Moufang loops with squares in the nucleus. The idea of constructing octonion loops by means of Fano plane, The proof of uniqueness of such a definition.
6. Pseudoautomorphisms and their applications in Moufang loops. Properties of associators and commutators in Moufang loops of nilpotency degree two. Quadratic forms and the construction of octonions. Code loops (the extent may differ from course to course). |
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Last update: prof. RNDr. Aleš Drápal, CSc., DSc. (01.06.2022)
Isotopy of loops and quasigroups and their interpretation by means of latin squares.
Important algebraic varieties of loops.
Constructions of various classes of loops.
Connections with projective planes.
Quasifields, nearfields and semifields.
Connections with group theory and cryptography.
More information may be retrieved from the exam requirements. |