Suzuki Group Sz Research Articles

On a group extension involving the Suzuki group Sz(8)

The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over {mathbb {F}}_{2}. Therefore a split extension group of the form 2^{12}{:}(Sz(8){:}3):= {overline{G}} exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of {overline{G}} by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of {overline{G}} is a 43 times 43 complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7.

A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

The spectrum ω(G) of a finite group G is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group G, there are four different primes such that ω(G) contains all their pairwise products but not a product of any three of these numbers, then G is nonsolvable. Using this result, we show that for q ⩾ 8 and q ≠ 32, the direct square Sz(q) × Sz(q) of the simple exceptional Suzuki group Sz(q) is uniquely characterized by its spectrum in the class of finite groups, while for Sz(32) × Sz(32), there are exactly four finite groups with the same spectrum.

A note on two families of 2-designs arose from Suzuki-Tits ovoid

In this note, we give a precise construction ofone of the families of 2-designs arose from studying ŕag-transitive 2-designs with parameters(v, k, λ) whose replication numbersrare coprime to λ. We show that for a given positive integer q=22n+1⩾8, there exists a 2-design with parameters (q2+ 1, q, q−1) and the replication numberq 2 admitting the Suzuki group Sz(q) asits automorphism group. We also construct a family of 2-designs with parameters (q2+ 1, q(q−1),(q−1)(q2−q−1)) and thereplication number q2(q−1) admitting the Suzuki groups Sz(q) astheir automorphism groups.

Formula omitted] designs admitting automorphism groups with socle Sz(q)

The classification of all block-transitive 2−(v,k,1) designs has been known as a long-standing and still open problem. This work is a contribution to the general program to classify the pairs (D,G) in which D is a 2−(v,k,1) design and G act as a point-primitive block-transitive automorphism group of D with a Suzuki group Sz(q) socle. We determine the point-stabilizer Gα ∩ Sz(q) and the block-stabilizer GB ∩ Sz(q), and the parameter of D is given.

Is the Suzuki group Sz(8) a subgroup of the Monster?

As a contribution to an eventual solution of the problem of the determination of the maximal subgroups of the Monster we show that there is no subgroup isomorphic to Sz ( 8 ) . This also completes the determination of exactly which simple groups are subgroups of which of the 26 sporadic simple groups. The proof is largely, though not entirely, computer-free.

The Depth of Subgroups of Suzuki Groups

We determine the combinatorial depth of certain subgroups of simple Suzuki groups Sz(q), among others, the depth of their maximal subgroups. We apply these results to determine the ordinary depth of these subgroups.

Intriguing sets of [formula omitted], q even

Infinite families of (q+1)-ovoids and (q2+1)-tight sets of the symplectic polar space W(5,q), q even, are constructed. The (q+1)-ovoids arise from relative hemisystems of the Hermitian surface H(3,q2) and from certain orbits of the Suzuki group Sz(q) in his projective 4-dimensional representation. The tight sets are closely related to the geometry of an ovoid of W(3,q). Other constructions of sporadic intriguing sets are also given.

Beauville surfaces and finite groups

Extending results of Bauer, Catanese and Grunewald, and of Fuertes and González-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L 2 ( q ) and SL 2 ( q ) for all prime powers q > 5 , and the Suzuki groups Sz ( 2 e ) and the Ree groups R ( 3 e ) for all odd e ⩾ 3 . We also show that L 2 ( q ) and SL 2 ( q ) admit strongly real Beauville structures, yielding real Beauville surfaces, for all q > 5 .

On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By $$Z^G_{p^\prime}$$ we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that $$Z^G_{p^\prime}$$ is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that $$Z^G_{p^\prime}$$ is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Kulshammer generalized this result to blocks with abelian defect groups. Here, we show that $$Z^G_{2^\prime}$$ is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore $$Z^{PSL(2,q)}_{p^\prime}$$ and $$Z^{PGL(2,q)}_{p^\prime}$$ are algebras for all primes p and all prime powers q. Furthermore we prove that $$Z^{Sz(q)}_{p^\prime}$$ is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|.

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2e or q=3e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.

An example of relations on the Ext-quiver for the Suzuki group Sz(8) in characteristic 2

Let G be the smallest Suzuki group Sz(8) and let F be an algebraically closed field of characteristic 2. The basic algebra of the group algebra of G over F is described by its Ext-quiver and a certain set of relations.

Characters of some finite groups of Lie type with a restriction containing a linear character once

Let G be a finite group of characteristic p of one of the types symplectic group Sp ( 4 , 2 f ) , Chevalley group G 2 ( q ) , Suzuki group Sz ( q ) or Ree group Re ( q ) . In this paper we show that if H is a Sylow p-subgroup of G then the restriction χ H of χ to H has at least one linear constituent of multiplicity one for certain irreducible characters χ of G.

Recognising the Suzuki groups in their natural representations

Under the assumption of a certain conjecture, for which there exists strong experimental evidence, we produce an efficient algorithm for constructive membership testing in the Suzuki groups Sz ( q ) , where q = 2 2 m + 1 for some m > 0 , in their natural representations of degree 4. It is a Las Vegas algorithm with running time O{} ( log ( q ) ) field operations, and a preprocessing step with running time O{} ( log ( q ) log log ( q ) ) field operations. The latter step needs an oracle for the discrete logarithm problem in F q . We also produce a recognition algorithm for Sz ( q ) = 〈 X 〉 . This is a Las Vegas algorithm with running time O{} ( | X | 2 ) field operations. Finally, we give a Las Vegas algorithm that, given 〈 X 〉 h = Sz ( q ) for some h ∈ GL ( 4 , q ) , finds some g such that 〈 X 〉 g = Sz ( q ) . The running time is O{} ( log ( q ) log log ( q ) + | X | ) field operations. Implementations of the algorithms are available for the computer system Magma.

Residual properties of the modular group and other free products

Using a probabilistic approach we establish new residual properties of the modular group PSL 2( Z) , and of more general free products. We prove that the modular group is residually in any infinite collection of finite simple groups not containing a Suzuki group Sz( q) or a 4-dimensional symplectic group PSp 4( q) with q a power of 2 or 3. This result is best possible, since the groups excluded are not quotients of the modular group. We also show that if S is a collection of classical groups of unbounded rank, then an arbitrary free product A∗B of nontrivial finite groups, not both 2-groups, is residually S, and prove results about free products A∗ Z .

Suzuki groups and automorphisms of finite linear spaces

We prove that no Suzuki group Sz( q) can act line-transitively on a non-trivial finite linear space.

Suzuki groups Sz(q) and 2-(v,k,1) block designs

It is proved that if D be a 2-(v,k,1) design with G⩽ Aut D block primitive then G does not have a Suzuki group Sz(q) as the socle.

Embeddings of Sz(32) in E 8 (5)

We show that the Suzuki group Sz(32) is a subgroup of E_8(5), and so is its automorphism group. Both are unique up to conjugacy in E_8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E_8(5).

A characterization of finite nonsimple groups by the set of orders of their elements

It is proved that the permutation wreath product H of a simple Suzuki group Sz(27) and a subgroup fo a symmetric group of degree 23, isomorphic to a Frobenius group of order 253, is (up to isomorphism) distinguished among all finite groups by the set of orders of its elements. Since H possesses a minimal normal subgroup N that contains an element of order equal to the exponent of N, this result furnishes a counterexample to one of the conjectures set forth by Shi [1]. In addition, we show that the direct square of a group Sz(27) is also distinguished by the set of orders of its elements.

Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, pv) and the projective special linear groups PSL(2, pv) are realized for allp, and the Suzuki groups Sz(22v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.

Supplement to The Structure of the Projective Indecomposable Modules of the Suzuki Group Sz(8) in Characteristic 2

Supplement to The Structure of the Projective Indecomposable Modules of the Suzuki Group Sz(8) in Characteristic 2