Simple Suzuki Groups Research Articles

Groups with the same character degrees as almost simple Suzuki groups

Let [Formula: see text] be a finite group and [Formula: see text] denote the set of complex irreducible character degrees of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group and [Formula: see text] is an almost simple group with socle [Formula: see text] such that [Formula: see text], then [Formula: see text] and [Formula: see text] is isomorphic to [Formula: see text].

TI subgroups and depth 3 subgroups in simple Suzuki groups

Abstract In this paper, we determine the TI subgroups of the simple Suzuki groups Sz ⁢ ( q ) \mathrm{Sz}(q) . More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively by Higman. These results were extended later by Goldschmidt, Shaw, Shult, Gross, Wilkens and Bryukhanova. As a corollary of our investigations, we get some interesting results for the Sylow 2-subgroups of Suzuki groups, as well. We relate this to an open problem on Suzuki 2-groups, and we ask a question concerning that. We also give some characterization of Suzuki groups.

THE SUBGROUP COMMUTATIVITY DEGREE OF FINITE -GROUPS

The subgroup commutativity degree of a group $G$ is the probability that two subgroups of $G$ commute, or equivalently that the product of two subgroups is again a subgroup. For the dihedral, quasi-dihedral and generalised quaternion groups (all of 2-power cardinality), the subgroup commutativity degree tends to 0 as the size of the group tends to infinity. This also holds for the family of projective special linear groups over fields of even characteristic and for the family of the simple Suzuki groups. In this short note, we show that the family of finite $P$-groups also has this property.

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.

The existence of minimal logarithmic signatures for the sporadic Suzuki and simple Suzuki groups

A logarithmic signature for a finite group G is a sequence [A1,? ,As] of subsets of G such that every element g?G can be uniquely written in the form g=g1?gs, where gi?Ai, 1≤i≤s. The aim of this paper is proving the existence of an MLS for the Suzuki simple groups Sz(22m+1), m>1, when 22m+1+2m+1+1 or 22m+1?2m+1+1 are primes. The existence of an MLS for untwisted group G2(4) and the sporadic Suzuki group Suz are also proved. As a consequence of our results, we prove that the simple groups have an MLS.

On the subgroup permutability degree of the simple Suzuki groups

We prove that the subgroup permutability degree of the simple Suzuki groups vanishes asymptotically. In the course of the proof we establish that the limit of the probability of a subgroup of $\Sz(q)$ being a 2-group is equal to 1.

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)|.