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Here “group” means additive abelian group. A compact group G contains δ–subgroups, that is, compact totally disconnected subgroups Δ such that G/Δ is a torus. The canonical subgroup Δ(G) of G that is the sum of all δ–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of Δ(G) when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following: Δ(G) contains tor(G), is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and G/Δ(G) is torsion-free and divisible; Δ(G) is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup Δ(G) appeared before in the literature as td(G) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups.

Let π be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a π - submaximal subgroup if there is a monomorphism ϕ : X → Y into a finite group Y such that X ϕ is subnormal in Y and H ϕ = K ∩ X ϕ for a π -maximal subgroup K of Y . In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the π -submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set π of primes, we obtain a description of the π -submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.

One pervasive facet of human interactions is the tendency to favor ingroups over outgroups. Remarkably, this tendency has been observed even when individuals are assigned to minimal groups based on arbitrary markers. Why is mere categorization into a minimal group sufficient to elicit some degree of ingroup favoritism? We consider several accounts that have been proposed in answer to this question and then test one particular account, which holds that ingroup favoritism reflects in part an abstract and early-emerging sociomoral expectation of ingroup support. In violation-of-expectation experiments with 17-mo-old infants, unfamiliar women were first identified (using novel labels) as belonging to the same group, to different groups, or to unspecified groups. Next, one woman needed instrumental assistance to achieve her goal, and another woman either provided the necessary assistance (help event) or chose not to do so (ignore event). When the two women belonged to the same group, infants looked significantly longer if shown the ignore as opposed to the help event; when the two women belonged to different groups or to unspecified groups, however, infants looked equally at the two events. Together, these results indicate that infants view helping as expected among individuals from the same group, but as optional otherwise. As such, the results demonstrate that from an early age, an abstract expectation of ingroup support contributes to ingroup favoritism in human interactions.

Abstract Let $$\\pi $$ π be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a $$\\pi $$ π -submaximal subgroup if there is a monomorphism $$\\phi :X\\rightarrow Y$$ ϕ:X→Y into a finite group Y such that $$X^\\phi $$ Xϕ is subnormal in Y and $$H^\\phi =K\\cap X^\\phi $$ Hϕ=K∩Xϕ for a $$\\pi $$ π -maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $$\\pi $$ π -submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set $$\\pi $$ π of primes, we obtain a description of the $$\\pi $$ π -submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.

People do not only feel guilty for transgressions that they are causally responsible for (i.e., personal guilt); they also feel guilty for transgressions committed by those they identify as in-group members (i.e., collective or group-based guilt). Decades of research using scenario-based imagination tasks and self-reported measures has shown that when reminded of transgressions committed by in-group members, people express guilt and are willing to make amends, even when they are not causally responsible for the transgressions. However, it remains elusive whether people genuinely experience guilt or simply display remorseful gestures deemed appropriate in those contexts. To resolve this puzzle, it is critical to closely examine the neurocognitive basis of group-based guilt and its relationship with personal guilt, a goal that self-reported measures alone cannot satisfactorily achieve. Here, we combined functional MRI with an interaction-based minimal group paradigm in which participants either directly caused harm to a group of victims (i.e., personal guilt), or observed in-group members cause harm to the victims (i.e., group-based guilt). In three experiments (N ​= ​90), we demonstrated and replicated that the perceived responsibility one shared with in-group members in transgression predicted both behavioral and neural manifestations of group-based guilt. Multivariate pattern analysis (MVPA) of the functional MRI data showed that group-based guilt recruited patterns of neural responses in anterior middle cingulate cortex that resembled personal guilt. These results have broadened our understanding of how group membership is integrated into the neurocognitive processes underlying social emotions. •An interaction-based minimal group paradigm was used to elicit group-based and personal guilt in laboratory settings.•Group-based guilt motivated reparation for transgression committed by in-group members.•Shared responsibility with in-group members in a transgression explained group-based guilt and reparation.•Group-based guilt was associated with activations in anterior middle cingulate cortex and right anterior insula.•Group-based guilt and personal guilt shared a similar neurocognitive representation.

I propose the need to establish a comparative study of societies, conceived of specificially here as bounded groups beyond a simple, immediate family that have the potential to endure for generations, whose constituent individuals recognize one another as members, and that maintain control over a physical space. This definition, with refinements and ramifications I explore, serves for cross-disciplinary research because it applies not just to nations but to diverse hunter–gatherer and tribal groups with a pedigree that likely traces back to the societies of our common ancestor with the chimpanzees. It also applies to groups among other species for which comparison to humans can be instructive. Notably, it describes societies in terms of shared group identification rather than social interactions. An expansive treatment of the topic is overdue given that the concept of a society (even the use of such synonyms as primate “troop”) has fallen out of favor among biologists, resulting in a semantic mess; whereas sociologists rarely consider societies beyond nations, and social psychologists predominantly focus on ethnicities and other component groups of societies. I examine the relevance of societies across realms of inquiry, discussing the ways member recognition is achieved; how societies compare to other organizational tiers; and their permeability, territoriality (allowing for mobile territories), relation to social networks and kinship, and impermanence. We have diverged from our ancestors in generating numerous affiliations within and between societies while straining the expectation of society memberships by assimilating diverse populations. Nevertheless, if, as I propose, societies were the first, and thereafter the primary, ingroups of prehistory, how we came to register society boundaries may be foundational to all human “groupiness.” A discipline-spanning approach to societies should further our understanding of what keeps societies together and what tears them apart.

We investigate the automorphism groups of ℵ0\\aleph _0-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable ℵ0\\aleph _0-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and calculate the WAP compactification in a number of examples.

We study several stronger versions of sensitivity for minimal group actions, including n -sensitivity, thick n -sensitivity and blockily thick n -sensitivity, and characterize them by the regionally proximal relation.

Empathic behavior and related neural processing are strongly modified by group membership. Shared neural circuits for the production and perception of facial emotional expressions represent mirror neuron mechanisms which play a pivotal role for empathy. In this study, we investigate the influence of group membership on mirror neuron mechanisms for emotional facial expressions. In a functional magnetic resonance imaging task, 178 healthy subjects perceived emotional and neutral facial expressions of artificial ingroup and outgroup members, displayed as 5 s video clips, and produced these facial expressions themselves. Before scanning, artificial group membership was manipulated ad-hoc through a minimal group paradigm. Shared neural activity for emotional facial expression production and perception was revealed in a large network with right-hemispheric preponderance encompassing motor mirror neuron regions, i.e., inferior frontal gyrus, supplementary motor area and middle temporal gyrus, in addition to limbic regions, i.e., amygdala, hippocampus, para-hippocampus, and insula. Within this network there was greater neural activation for ingroup compared to outgroup members in temporal poles, amygdalae, the left insula, the left inferior frontal gyrus, and the inferior and middle temporal gyrus, the right hippocampus and parahippocampus. We validate and extend knowledge on brain regions with mirror neuron properties. Most crucially, we provide evidence for the influence of group membership on regions within the mirror neuron system, indicating more neural resonance (mirroring) for ingroup facial emotional expressions. •There is enhanced neural resonance (mirroring) for ingroup facial emotions.•Such ingroup bias affects amygdala, temporal pole, and hippocampus activation.•Ad-hoc created minimal group membership sufficiently induces this ingroup bias.•Knowledge about brain regions with mirror neuron properties is extended.

The Integrated self-categorization model of autism (ISCA) argues that a self-categorization dysfunction could be the link between some of the disparate features of ASD. To the extent that this is true, any social psychological phenomena arising from self-categorization should be impaired in autistic people. Based on this premise, we investigated whether ingroup favouritism within the minimal group paradigm is reduced to the extent that individuals possess autistic traits. Results indicated that participants with a high proportion of autistic traits showed less ingroup favouritism, and that this was due to a decreased tendency for self-categorization. By providing evidence of the disruption of self-categorization in ASD, these findings lend support to ISCA and raise important issues for existing accounts of the disorder.