Multiplier, also called multiplier, is a special classAutomorphism。Set D togroupOne of G (v, k, λ)Difference set, the operation of G is additive, α is one of GAutomorphism。If there is a, b ∈ G, let Dα=A+D+b, then α is called DMultiplier。When α is zero, α is calledRight multiplier；When G isAbelian groupIf there is an integer m and α is a mapping x → mx, then α is called aNumerical multiplier, sometimes called m, is a numerical multiplier.[1]
All multipliers of 1. D become onegroup, and all right multipliers aresubgroup。
2. When G isAbelian groupAll multipliers areRight multiplier；When G isCyclic groupAll multipliers areNumerical multiplier。
3. When D is an Abelian difference set, a multiplier of D must fix a translation of D.Using this property and multiplier theorem (see below), we can construct some difference sets and prove the nonexistence of some difference sets.[1]
give an example
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For example, {1,3,9,5,4} is Zeleven(11,5,2) difference set in, m=3 is its numerical multiplier.[1]
Multiplier theorem
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Multiplier theorem is a theorem used to judge the existence of difference set multipliers.The multiplier theorem has many forms, and the following multiplier theorem is also called the second multiplier theorem.
Second multiplier theoremLet D be the (v, k, λ) difference set of Abelian group G of order v, m be a coprime factor of n=k - λ with v, and m>λ.If the integer t and v are coprime, so that there is a corresponding nonnegative integer f for each prime factor p of m, it is suitable for t ∨ pf(mod v *), where v * is the exponent of G, even if xe=1 The automorphism x → x of G is the smallest positive integer e for all x in GtIs the numerical multiplier of D.[1]
This theorem was obtained by Mann (H.B.) in 1965.When G is a cyclic group and λ=1, the earlier form is obtained by Hall M.Jr.Since the multiplier of Abel difference set D must be fixed to some translation of D, some difference sets can be made or the difference sets of some parameters can be proved not to exist by the multiplier theorem.For example, (11,5,2) cyclic difference set can be made as follows: if such difference set exists, then 3 is a numerical multiplier of D; if 3 is fixed, then D must be a cyclic group ZelevenThe union of some orbits of the elements of Z under the action of automorphism x → 3x, and ZelevenThe element orbits of are {0}, {1,3,9,5,4} and {2,6,7,10,8}.So both orbits are Zeleven(11,5,2) difference set in.For another example, if there is a cyclic (31,10,3) difference set D, then 7 should be the multiplier of D. It is better to let 7 fix D.However, under the action of automorphism x → 7x, Zthirty-oneIt is divided into three element orbits with lengths of 1,15 and 15, which means that D does not exist.[1]
In the second multiplier theorem, take m as the prime number p.Special cases of the theorem that can be obtained (called the first multiplier theorem):
First multiplier theoremLet D be a (v, k, λ)AbelDifference set, p is prime, p | n, p | v is not true, if p>λ, then p is a multiplier of D.[1]
The proof of this theorem depends on the condition p>λ, but in fact, for every knownAbelThe difference set must be a multiplier of the difference set as long as the prime p is a factor of n and does not divide v by an integer.Therefore, people guess that the first multiplier theorem is still true after removing the condition p>λ.This conjecture is calledMultiplier conjecture。
The multiplier theorem shows that the factor of n is an important source of multipliers.But this is not the only source.For example, 11 is a numerical multiplier of (21,5,1) cyclic difference D={3,6,7,12,14}, while 11 is not a factor of n=4.When a numerical multiplier is not a factor of n, it is calledExtra multiplier。It is known that some numbers cannot be additional multipliers of difference sets.For example, 2 cannot be an additional multiplier of Abelian difference set, and v-1 cannot be an additional multiplier of any (v, k, λ) cyclic difference set.[1]
