Automorphism Groups Of (v, K,2) Symmetric Designs With Order At Most25

There are deep relationships between the combinatorial designs and group theory.We have known that the automorphism groups of many2-(v, k, λ) symmetric designswith specific parameters v, k and λ. In recent years, the attention has turned to theautomorphism groups of symmetric designs with order at most25.In2009, A. Abdollahi, H.R. Maimani and R. Torabi obtained the order of the fullautomorphism group of the symmetric2-(81,16,3) design. In the same year, M. Alaeiyanand R. Safakish finished the research of the order of the full automorphism group ofsymmetric2-(121,16,2) design.By the methods of these papers and other new methods, we obtain the orders ofthe full automorphism groups of symmetric2-(154,18,2),2-(191,20,2),2-(211,21,2),2-(301,25,2) and2-(352,27,2) designs. In other words, we finish the study of the auto-morphism groups of symmetric designs with order at most25and λ=2.The main structure of the thesis is as follows:In Chapter I, we first introduce our research background, including the history andresearch situation on group and design theory. Then we give some elementary conceptsand results which will be used in this thesis.In Chapter II, we discuss the order of the automorphism group of the symmetric2-(154,18,2) design, and prove the following theorem:Theorem2.1Let D be a symmetric2-(154,18,2) design, the order of the fullautomorphism group is|Aut(D)|=2α3β5γ7δ11r13s17t, where r, s, t∈{0,1}, γ∈{0,1,2},and α, β, δ are non-negative integers.In Chapter III, we discuss the order of the automorphism group of the symmetric2-(191,20,2) design, and prove the following theorem:Theorem3.1Let D be a symmetric2-(191,20,2) design, the order of the fullautomorphism group is|Aut(D)|=2α3β5r17t19u191v, where t, u, v∈{0,1}, and α, β, rare non-negative integers.From Chapter IV to Chapter VI, we discuss the order of the automorphism groupof the symmetric2-(211,21,2),2-(301,25,2),2-(352,27,2) design respectively, and provethe following theorems:Theorem4.1Let D be symmetric2-(211,21,2) design, the order of the full auto-morphism group is|Aut(D)|=2α3β5γ7r17s19t211u, where γ, r, s, t, u∈{0,1}, and α, βare non-negative integer. Theorem5.1Let D be a symmetric2-(301,25,2) design, the order of the full au-tomorphism group is|Aut(D)|=2α3β5γ7δ11r23t301u, where r, t, u∈{0,1}, and α, β, γ, δare non-negative integers.Theorem6.1Let D be a symmetric2-(352,27,2) design, the order of the fullautomorphism group is|Aut(D)|=2α3β5γ7δ11r13s23t, where δ, s, t∈{0,1}, δ∈{0,1,2}and α, β, r are non-negative integers.