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A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

In this article we improve the known uniform bound for subgroup growth of Chevalley groups G(&#120125p[[t]]). We introduce a new parameter, the ridgeline number v(G), and give new bounds for the subgroup growth of G(&#120125p[[t]]) expressed through v(G). We achieve this by deriving a new estimate for the codimension of [U, V ] where U and V are vector subspaces in the Lie algebra of G. [ABSTRACT FROM AUTHOR]

In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured. [ABSTRACT FROM AUTHOR]

This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in PGL n ( F ) for n ≤ 7 which are Hurwitz, where F is an algebraically closed field. For the groups G 2 ( q ) , q ≥ 5 , and the Janko groups J 1 and J 2 it provides explicit ( 2 , 3 , 7 ) -generators. [ABSTRACT FROM AUTHOR]

We determine all saturated fusion systems ℱ on many 2-groups P with exactly three involutions. For this we use Janko's classification of these groups ([Groups of Prime Power Order. Vol. 1, Walter de Gruyter, Berlin, 2008, Section 82]) as a guide. On the way we describe all such P which contain an ℱ-essential subgroup isomorphic to or to the Sylow 2-subgroup of PSU(3,4). Furthermore, we classify all saturated fusion systems on these P. This contributes to the outstanding classification of fusion systems on 2-groups of 2-rank 2. [ABSTRACT FROM AUTHOR]

For q ∈ {7, 8, 9, 11, 13, 16}, we consider the primitive actions of L2(q) and use Key-Moori Method 1 as described in [Codes, designs and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002) 143-159., Correction to: "Codes, designs and graphs from the Janko groups J1 and J2" [J. Combin. Math. Combin. Comput. 40 (2002) 143-159], J. Combin. Math. Combin. Comput., 64 (2008) 153.] to construct designs from the orbits of the point stabilisers and from any union of these orbits. We also use Key-Moori Method 2 (see Information security, coding theory and related combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS Amsterdam, 29 (2011) 202-230.) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups. The incidence matrices of these designs are then used to generate associated binary codes. The full automorphisms of these designs and codes are also determined. [ABSTRACT FROM AUTHOR]

In this article, we classify p-groups whose non-normal subgroups have index p in their normalizers. This answers a problem posed by Berkovich and Janko. [ABSTRACT FROM AUTHOR]

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $ J_1$ and $ J_3$ [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the decomposition numbers of the non-principal blocks of the largest Janko group [Z. Janko, A new finite simple group of order 86,775,571,046,077,562,880 which possesses and the full covering group of as subgroups, J. Algebra 42 (1976) 564-596] for characteristic 3 are determined up to some unknown parameters. We are also concerned with the decomposition numbers of maximal 2-local subgroups of in odd characteristics. [Copyright &y& Elsevier]

Abstract: There are only finitely many locally projective regular polytopes of type {5, 3, 5}. They are covered by a locally spherical polytope whose automorphism group is , where is the first Janko group, of order 175560, and is the projective special linear group of order 3420. This polytope is minimal, in the sense that any other polytope that covers all locally projective polytopes of type {5, 3, 5} must in turn cover this one. [Copyright &y& Elsevier]

Abstract: In 1955 Hall and Paige conjectured that a finite group is admissible, i.e., admits complete mappings, if its Sylow 2-subgroup is trivial or noncyclic. In a recent paper, Wilcox proved that any minimal counterexample to this conjecture must be simple, and further, must be either the Tits group or a sporadic simple group. In this paper we improve on this result by proving that the fourth Janko group is the only possible minimal counterexample to this conjecture: John Bray reports having proved that this group is also not a counterexample, thus completing a proof of the Hall–Paige conjecture. [Copyright &y& Elsevier]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]

Abstract: We construct self-dual codes of length 100 invariant under the Hall–Janko group . We also construct a 2- design with the automorphism group , which is the extension of by its outer automorphism. [Copyright &y& Elsevier]

Abstract: Uno''s invariant conjecture for the Chevalley groups has been verified when the characteristic of the modular representation is distinct from the defining characteristic of the groups. Thus Dade''s reductive conjecture and the Isaacs–Navarro conjecture both hold for in non-defining characteristics. [Copyright &y& Elsevier]

In this paper we give a self-contained existence and uniqueness proof for the sporadic O'Nan group ON by showing that it is uniquely determined up to isomorphism by the centralizer H of a 2-central involution z. We establish for such a simple group G a presentation in terms of generators and defining relations and a faithful permutation representation of degree 2.624.832 with a uniquely determined stabilizer isomorphic to the small sporadic Janko group J1. We also calculate its character table by new methods and determine a system of representatives of the conjugacy classes of G. [ABSTRACT FROM AUTHOR]

A partition for the elements of prime-power order in a finite group G is a family of subgroups with the property that every non-identity element of prime-power order lies in exactly one subgroup of the family. The main result of this paper is a classification of the finite simple groups which have such a partition. We also establish a connection between this concept and the class of permutation groups all of whose elements of prime-power order have the same number of fixed points. [ABSTRACT FROM AUTHOR]

We demonstrate how readily a definition of the largest Janko group J4 follows from a primitive action of the Mathieu group M24 by exhibiting J4 as an image of the progenitor 2*3795:M24. This symmetric presentation is converted into an ordinary presentation on three generators. [ABSTRACT FROM AUTHOR]