Search

Presenting sporadic simple groups as an automorphism groups of designs and graphs is an exciting field in finite group theory. In this paper, with two different methods, we present some new resolvable simple 3-designs with Higman-Sims sporadic simple group HS as the full automorphism group. Also, we classify all block-transitive self-orthogonal designs on 176 points with even block size that admit sporadic simple group HS as an automorphism group. Furthermore, with these methods we construct some new resolvable 3-designs on 36, 40, 120 and 176 points. [ABSTRACT FROM AUTHOR]

In this paper, we determine all simple block-transitive 3-designs on 65 points with block stabilizer of order greater than 2, admitting the 2-transitive action of the group S z (8). As a result, we find 23 3-designs on 65 points that have S z (8) as an automorphism group. Also we construct four 3-(3 6 , 1 8 , λ 1 ) designs for λ 1 ∈ { 7 6 8 , 5 1 8 4 , 6 9 1 2 , 9 2 1 6 } and two 3-(3 6 , 1 7 , λ 2 ) designs for λ 2 ∈ { 2 8 8 0 , 8 6 4 0 }. All of these designs have S 6 (2) as the full automorphism group. Some of these designs are flag and anti-flag-transitive. [ABSTRACT FROM AUTHOR]

A group G is said to have an exact factorization if there exist proper subgroups Ai for i = 1 , 2 , ... , n such that G = A 1 A 2 ... A n and | G | = | A 1 |. | A 2 | ... | A n | . The number n is called length of this factorization. An exact factorization of length n is called minimal length if there is no exact factorization of length less than n. In this paper, up to conjugacy, we determine all minimal length exact factorizations of sporadic simple groups J 1 , M 22 , J 2 , and HS . With our results, the classification of minimal length exact factorizations of sporadic simple groups and their automorphism groups of the order less than or equal to | M 24 | except for J 3 are completed. Also, we show that the group J 3 has no exact triple factorization and the minimal length for an exact factorization of the group McL is 4. [ABSTRACT FROM AUTHOR]

A finite group G has uniform spread k if there exists a fixed conjugacy class C of elements in G with the property that for any k nontrivial elements s1, s2, ..., sk in G there exists y ∊ C such that G = (s1, y) for i = 1, 2, ..., k. Further, the exact uniform spread of G is the largest k such that G has the uniform spread k. In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups. [ABSTRACT FROM AUTHOR]

Presenting sporadic simple groups as automorphism group of designs and graphs is an exciting field in simple group theory. In this paper, we present five simple 3-designs with Janko sporadic group J 2 automorphism. Also, we classify all block-transitive self-orthogonal designs and block-transitive t -designs for t ≥ 2 on 100 points with block stabilizer of order greater than four that admit sporadic group J 2 as automorphism group. Moreover, we determine all flag-transitive designs on 100 points with automorphism group J 2. [ABSTRACT FROM AUTHOR]