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Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^{\Omega} (G)$ of $D(G)$ generated by relative syzygies $\Omega_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions defined on the $p$-subgroups of $G$, there are natural generators $\omega_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $\Psi_G:C(G,p)\to D^\Omega(G)$ that sends $\omega_X$ to $\Omega_X$. The main theorem of the paper is the verification that the subgroup of $C(G,p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $\Psi_G$., Comment: Minor revision, 40 pages

We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system. We also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system.

Cataloged from PDF version of article. Thesis (M.S.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2020. Includes bibliographical references (leave 91-93). Of the 26 sporadic finite simple groups, 5 were discovered by E. Mathieu in 1861 and 1873 [1], [2]. These Mathieu groups are the focus of this thesis, where we will prove their simplicity using elementary methods. E. Witt [5] realized a connection between the Mathieu groups and certain combinatorial structures known as Steiner systems. We will follow his construction to define the Mathieu groups as the automorphism groups of certain Steiner systems. Much of the work of the thesis lies in the construction of these Steiner systems, which we achieve by using both methods from finite geometry and the theory of Golay codes. by Mete Han Karakaş M.S.

Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is invariant under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact form a semicharacteristic biset for the fusion system on S induced by G. The parameterization of such semicharacteristic bisets can then be applied to relate the module structure and defect theory of B., Comment: 8 pages

Füzyon sistemleri sonlu bir grubun p-lokal yapılarını i¸ceren kategorilerdir. İkili kümeler iki grubun uyumlu etkisine sahip kümelerdir. Biz bu tezde, füzyon sistemleri ve ikili kümeler arasındaki ilişkiyi araştıracağız.Bir sonlu grup ile Sylow-p altgrubu arasındaki ilişkiyi taklit eden füzyon sistemlerine doygun füzyon sistemi denir. Eğer S sonlu bir grup olan G'nin Sylow p-altgrubu ise, (S,S)-ikili kümesi olarak düşünülebilen G grubu kendine has özelliklere sahiptir ki bu özellikler, onu G'nin p-füzyonunda karakteristik ikili bir küme yapar. Bu iki kavram şu şekilde ilişkilidir: Bir füzyon sistemi ancak ve ancak karakteristik bir ikili kümeye sahipse doygundur. Bu sonucu, [1] ve [2]'deki çalışmaları takip ederek ispatlıyoruz.Füzyon sistemleri sonlu grup teorisindeki normalleyici ve sabitleyici kavramlarını taklit ederek kendine has normalleyici ve sabitleyici alt sistemlere sahiptir. Bu tez, Puig tarafından [2]'de kanıtlanan doygun füzyon sistemlerinin normalleyici ve sabitleyici füzyon alt sistemlerinin de doygun olması sonucunu, [3]'deki bir ispatı gözden geçirirek tekrar inceliyor. Bu sonuç karakteristik ikili kümelerin varlığından ve füzyon sistemlerini arasındaki ilişkiden gelmektedir. Fusion systems are categories that contain the p-local structure of a finite group. Bisets are sets endowed with two coherent group actions. We investigate the relation between fusion systems and bisets in this thesis.Fusion systems that mimic the inclusion of a Sylow p-subgroup of a finite group are called saturated. Similarly, if S is a Sylow p-subgroup of G, then G regarded as an (S, S)-biset has special properties, which make it a characteristic biset for the p-fusion of G. These two concepts are linked in that a fusion system is saturated if and only if it has a characteristic biset. We give a proof for this result by following the work in [1] and [2].Fusion systems have a notion of normalizer and centralizer subsystems, mimicking the notion for finite group theory. This thesis reviews a proof by Gelvin and Reeh [3] of a result of Puig [2] asserting that normalizer and centralizer fusion subsystems of a saturated fusion system are saturated. This result comes from the connection between saturation of fusion systems and the existence of characteristic bisets. 55