On the modular isomorphism problem for groups of class 3 and obelisks

The Modular Isomorphism Problem for small groups – revisiting Eick's algorithm

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

On monomial codes in modular group algebras

Chapter 12 - Mapping Class Groups

Purpose: Cooperative learning structures allow diverse groups of students to learn together while developing higher-order thinking skills, yet there are concerns that not all students’ voices are being heard. The intent of this study was to discover the relationship between the instructional methodologies one teacher used to foster the development of group skills in students and the ways small discussion groups actually functioned. This study explored issues of power and status on students’ learning opportunities in middle school discussion groups. Method: This qualitative study was conducted under an ethnographic lens using a teacher-as-researcher approach. Data were collected over one school year in a sixth grade classroom from two different class groups. Data sources included field notes, audio and video-recordings, and teacher observations of groups. A sociometric device was used to measure the peer status of the forty-eight student participants. Thirty-one recordings of student discussion groups were transcribed and analyzed using Fairclough’s (2004) methods of critical discourse analysis. Transcripts were coded for Mercer’s (1995) three types of talk: disputational, cumulative, and exploratory. Student participation was measured as a percentage of total group discussion. Students’ peer status, gender, and participation rates were compared. Analysis/Results: In one class group, a relationship was found between gender, status, and participation. Students gradually adopted the genre, discourse, and style of academic discussions. Students with low peer status increased participation rates over time, and students with high peer status decreased participation rates over time. Five student participation patterns emerged from the data: facilitating, contributing, dependent, silent, and distracting. The percentage of total group talk spent on disputational and organizational talk decreased over time while cumulative and exploratory talk increased. Discussion: Peer status effects were found in one class group, but these effects decreased as students developed academic discussion skills. Students’ high-level thinking and talking increased over the school year. Connections can be made between instruction of academic discourse and student success in discussion groups. Examination of the five patterns of student participation provides insight into how to foster high-level discussion skills in students. The generalizability of this study to other educational settings is addressed.

Automorphism groups of simple Kokoris algebras

In this article, we investigate the powerful nilpotency class of powerfully nilpotent groups of standard nilpotency class 2. We outline the process of collecting data using the computer algebra system GAP, formulating a conjecture based on the data, and finally we prove the conjecture. In particular, we prove that for a powerfully nilpotent group of nilpotency class 2 and order pn , where p is an odd prime, the powerful nilpotency class of G is at most the integer part of . We also identify and explain what this means in terms of the powerful coclass of the group.

Let [math] , for [math] , be the moduli stack of [math] –pointed, genus [math] , stable complex curves of compact type. Various characterizations and properties are obtained of both the topological and algebraic fundamental groups of the stack [math] . For instance we show that the topological fundamental groups are linear, extending to all [math] previous results of Morita and Hain for [math] and [math] .\n¶ Let [math] , for [math] , be the Teichmüller group associated with a compact Riemann surface of genus [math] with [math] points removed [math] , ie the group of homotopy classes of diffeomorphisms of [math] which preserve the orientation of [math] and a given order of its punctures. Let [math] be the normal subgroup of [math] generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on [math] . The above theory yields a characterization of [math] for all [math] , improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].\n¶ The Torelli group [math] is the kernel of the natural representation [math] . The abelianization of the Torelli group [math] is determined for all [math] and [math] , thus completing classical results of Johnson [Topology 24 (1985) 127-144] and Mess [Topology 31 (1992) 775-790] for closed and one-punctured surfaces.\n¶ We also prove that a connected finite étale cover [math] of [math] , for [math] , has a Deligne–Mumford compactification [math] with finite fundamental group. This implies that, for [math] , any finite index subgroup of [math] containing [math] has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

Let 𝐹 be a field of odd characteristic and 𝐺 a group for which the group algebra F ⁢ G FG is modular. In this paper, necessary and sufficient conditions are presented for the commutator subgroup of the unit group of F ⁢ G FG to be nilpotent of class at most 2.

Modular group algebras with centrally metabelian unit groups

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text] for [Formula: see text] and [Formula: see text]. Group algebras of upper Lie nilpotency index [Formula: see text] for [Formula: see text], have already been characterized completely.

Let K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

Some finitely presented nilpotent-by-abelian groups

In this paper, we classify the modular group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] having upper Lie nilpotency index [Formula: see text]. The group algebra [Formula: see text] with [Formula: see text] has already been described.

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only ifwhere c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.