On a two-fold cover 2.(2⁶˙G₂(2)) of a maximal subgroup of Rudvalis group Ru

The Schur multiplier M(Ḡ1) ≅4 of the maximal subgroup Ḡ1 = 2⁶˙G₂(2)of the Rudvalis sporadic simple group Ru is a cyclic group of order 4. Hence a full representative group R of the type R = 4.(2⁶˙G₂(2)) exists for Ḡ1. Furthermore, Ḡ1 will have four sets IrrProj(Ḡ1;αi) of irreducible projective characters, where the associated factor sets α1, α2, α3 and α4, have orders of 1, 2, 4 and 4, respectively. In this paper, we will deal with a 2-fold cover 2. Ḡ1 of Ḡ1 which can be treated as a non-split extension of the form Ḡ = 27˙G2(2). The ordinary character table of Ḡ will be computed using the technique of the so-called Fischer matrices. Routines written in the computer algebra system GAP will be presented to compute the conjugacy classes and Fischer matrices of Ḡ and as well as the sizes of the sets |IrrProj(Hi; αi)| associated with each inertia factor Hi. From the ordinary irreducible characters Irr(Ḡ) of Ḡ, the set IrrProj(Ḡ1; α2) of irreducible projective characters of Ḡ1 with factor set α2 such that α22= 1, can be obtained.

Operator System Structures and Extensions of Schur Multipliers

For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $\varphi $ and, under some richness conditions, characterise its affirmative solution in terms of the equality between the canonical and the maximal dual operator $\mathcal{A}$-system structures on an operator system naturally associated with the domain of $\varphi $.

Schur multipliers of special 𝑝-groups of rank 2

Let G be a special p-group with center of order {p^{2}}. Berkovich and Janko asked to find the Schur multiplier of G in [Y. Berkovich and Z. Janko, Groups of Prime Power Order. Volume 3, De Gruyter Exp. Math. 56, Walter de Gruyter, Berlin, 2011; Problem 2027]. In this article, we answer this question by explicitly computing the Schur multiplier of these groups.