Remarks on Singer Cyclic Groups and Their Normalizers

Cossidente, Antonio; Resmini, Marialuisa J. de

doi:10.1023/b:desi.0000029214.50635.17

Cited by 21 publications

(55 citation statements)

References 7 publications

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“…Let M := x i | 0 ≤ i < q n −1 q−1 . The first two statements in the remark below follow from results in [2,7,25]. The third statement follows from the second statement and the facts that Z G ≤ stab G ( 1 ) and M ∈ lt (H, Z G ).…”

Section: Mlss For G L N (Q) and P G L N (Q)mentioning

confidence: 81%

Minimal logarithmic signatures for finite groups of Lie type

Singhi¹,

Singhi²,

Magliveras³

2010

Des. Codes Cryptogr.

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A logarithmic signature (LS) for a finite group G is an ordered tuple α = [A 1 , A 2 , . . . , A n ] of subsets A i of G, such that every element g ∈ G can be expressed uniquely as a product g = a 1 a 2 · · · a n , where a i ∈ A i . The length of an LS α is defined to be l(α) = n i=1 |A i |. It can be easily seen that for a group G of order k j=1 p j m j , the length of any LS α for G, satisfies, l(α) ≥ k j=1 m j p j . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002. The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families P SL n (q) and P Sp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.

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Section: Mlss For G L N (Q) and P G L N (Q)mentioning

confidence: 81%

Minimal logarithmic signatures for finite groups of Lie type

Singhi¹,

Singhi²,

Magliveras³

2010

Des. Codes Cryptogr.

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“…As described above, since the number of non-zero isotropic points in P (V ) with respect to the quadratic space (V, Q) are (q m + 1)(q m−1 − 1)/(q − 1), therefore, we need to construct a cyclic group A of order q m + 1, which must be sharply transitive on a partial spread S and a cyclic set B of cardinality (q m−1 − 1)/(q − 1), which must be sharply transitive on the projective subspace of P (V ). Then, we take advantage of the Lemma 3, Lemma 5 and Lemma 6 for proving the existence of MLSs for O [19] . Then b 1 ∈ GL 2m (q) can be well defined as follows: Also, we observe that the group A 1 is sharply transitive on S We at first consider O + 2m (q).…”

Section: Lemma 4 [62123] If G Is a Solvable Group Then G Has An Mlsmentioning

confidence: 99%

Minimal logarithmic signatures for one type of classical groups

Hong

Wang

Ahmad

et al. 2016

AAECC

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As a special type of factorization of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like M ST 1 , M ST 2 and M ST 3 . An LS with the shortest length, called a minimal logarithmic signature (MLS), is even desirable for cryptographic applications. The MLS conjecture states that every finite simple group has an MLS. Recently, the conjecture has been shown to be true for general linear groups GL n (q), special linear groups SL n (q), and symplectic groups Sp n (q) with q a power of primes and for orthogonal groups O n (q) with q as a power of 2. In this paper, we present new constructions of minimal logarithmic signatures for the orthogonal group O n (q) and SO n (q) with q as a power of odd primes. Furthermore, we give constructions of MLSs for a type of classical groups -projective commutator subgroup P Ω n (q).

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“…If in addition, the group G preserves a product decomposition, then the same decomposition is preserved by G (2) . Thus, in this case, the form of this group can be found by means of the classification of 3/2-transitive imprimitive linear groups given in [8].…”

Section: Theorem 32 Let G ≤ Sym(v ) Be a Uniprimitive 3/2-transitivementioning

confidence: 99%

“…Then C = Inv(Γ ) and so Aut(C) = Γ (2) . On the other hand, it is easy to see that the orbits of the group Γ v other than {v} all have the same size |K|.…”

Section: Lemma 33 If the Scheme C Is Nontrivial Then T Is A Charactmentioning

confidence: 99%

“…(1) Γ has a normal subgroup Γ isomorphic to one of the classical groups SL(n, q ), Sp(n, q ), SU(n, q 1/2 ), or Ω (n, q ), where r divides the order of Γ , q is the order of a subfield of the ground field, and ∈ {•, +, −}, (2) Γ ≤ GL(n/m, q m ) · m, and the number r divides the order of the group Γ ∩ GL(n/m, q m ), where m is a divisor of n other than 1, (3) (q, n) = (2, 4) or (2,6), where GL(n/m, q m ) · m is the general linear group GL(n/m, q m ) embedded to GL(n, q) and extended by the group of automorphisms of the field extension GF(q m )/ GF(q). However, the case (3) does not arise by the hypothesis of the theorem.…”

Section: Theorem 34 Under the Conditions Of Theorem 11 Denote By Gmentioning

confidence: 99%

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On cyclotomic schemes over finite near-fields

2007

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We introduce a concept of cyclotomic association scheme over a finite near-field K. It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V ), where V is the linear space associated with K. A sufficient condition on a cyclotomic scheme C that guarantee the inclusion Aut(C) ≤ A L(1, F), where F is a finite field with |K| elements, is given.

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Remarks on Singer Cyclic Groups and Their Normalizers

Cited by 21 publications

References 7 publications

Minimal logarithmic signatures for finite groups of Lie type

Minimal logarithmic signatures for finite groups of Lie type

Minimal logarithmic signatures for one type of classical groups

On cyclotomic schemes over finite near-fields

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