The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

PROJECTIVE CHARACTERS WITH PRIME POWER DEGREES

We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

The smooth surfaces in ℙ4 with few apparent triple points

We classify smooth complex projective surfaces in [Formula: see text] with [Formula: see text] apparent triple points, thus recovering and extending the results of Ascione [Sulle superficie immerse in un [Formula: see text], le cui trisecanti costituiscono complessi di [Formula: see text] ordine, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.[Formula: see text]5[Formula: see text] 6 (1897) 162–169] and Severi [Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901) 33–51] for [Formula: see text], Marletta [Le superficie generali dell’ [Formula: see text] dotate di due punti tripli apparenti, Rend. Circ. Mat. Palermo 34 (1912) 179–186] for [Formula: see text], and Aure [The smooth surfaces in [Formula: see text] without apparent triple points, Duke Math. J. 57 (1988) 423–430] for [Formula: see text]. This is done thanks to a new projective character that can be introduced as a consequence of the main result of [K. Ranestad, On smooth plane curve fibrations in [Formula: see text], in Geometry of Complex Projective Varieties, Sem. Conf., Vol. 9 (Mediterranean, 1993), pp. 243–255; J. C. Sierra and A. L. Tironi, Some remarks on surfaces in [Formula: see text] containing a family of plane curves, J. Pure Appl. Algebra 209 (2007) 361–369; V. Beorchia and G. Sacchiero, Surfaces in [Formula: see text] with a family of plane curves, J. Pure Appl. Algebra 213 (2009) 1750–1755]. Going a bit further, we obtain some bounds on the Euler characteristic [Formula: see text] in terms of the degree [Formula: see text] and the sectional genus [Formula: see text] of a smooth surface in [Formula: see text]. For [Formula: see text], these bounds were first obtained in [A. B. Aure and K. Ranestad, The smooth surfaces of degree [Formula: see text] in [Formula: see text], in Complex Projective Geometry, London Mathematical Society Lecture Note Series, Vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 32–46; K. Ranestad, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Ph.D. thesis, Oslo (1988); S. Popescu, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Dissertation, Saarbrücken (1993)]. Here we give a different argument based on liaison that works also for [Formula: see text] and that allows us to determine the triples [Formula: see text] of the smooth surfaces with [Formula: see text] apparent triple points.

Reconceptualización de la identidad personal y educación para la autodeterminación posible

RESUMEN: En este artículo, procurando superar las tensiones dialécticas entre las ideas modernas y posmodernas, se analiza la identidad como construcción de significado en interacción y desde la narrativa personal. En este sentido, enfatizamos, dentro de la condición esencialmente dinámica de la identidad, su carácter proyectivo como «personalidad escogida». La identidad personal es una parte de la personalidad, esa parte que podemos imaginar, inventar, crear y escoger, con nuestras posibilidades y limitaciones individuales y situacionales, por nosotros mismos. Nuestra capacidad de elegir, de decidir, es ciertamente limitada. No obstante, en este ámbito se define la libertad humana, la cual concebimos aquí como nuestra capacidad de autodeterminación posible. Consecuentemente, refiriéndonos al propio sujeto, delimitamos los principales dominios formativos para la autodeterminación posible, desde la que cabe desplegar el descubrimiento de las estrategias educativas pertinentes para la construcción autónoma de la identidad.ABSTRACT: While attempting to overcome the dialectic tensions existing between modern and post-modern ideas, this article analyses identity as the construction of meaning in interaction and from the personal narrative. In this regard, we stress, within the essentially dynamic condition of personality, its projective character as the «chosen personality». Personal identity is one part of personality; this is the part that we ourselves, within our individual and situational possibilities and limitations, can imagine, invent, create and choose. Our capacity to choose, to decide is evidently limited yet within this restricted ambit human liberty, something that we regard here as being our possible capacity for self-determination, is defined. As a result, and referring to the subject him or herself, we define the principle formative ambits for this possible self-determination and thence proceed to discover the educational strategies relevant to the autonomous construction of the identity.SOMMAIRE: Dans cet article, en essayant de dépasser les tensions dialectiques entre les idées modernes et postmodernes, est analysée l'identité comme construction de signifié en interaction et depuis la narration personnelle. Dans ce sens, nous insistons, dans la condition essentiellement dynamique de l'identité, sur son caractère projectif comme «personnalité choisie». L'identité personnelle est une partie de la personnalité, cette partie que nous pouvons imaginer, inventer, créer et choisir, avec nos possibilités et limitations individuelles et situationnelles, pour nous-mêmes. Notre capacité de choix, de décision, est certes limitée. Cependant, dans ce cadre se définit la liberté de l'homme, que nous concevons ici comme notre capacité d'autodétermination possible. En conséquence, en nous rapportant au propre sujet, nous délimitons les principaux cadres formateurs pour l'autodétermination possible, depuis laquelle il y a lieu de déployer la découverte des stratégies éducatives pertinentes pour la construction autonome de l'identité.

THE BAD BEHAVIOR OF REPRESENTATION GROUPS

Let (H, A) be a primitive central extension of a finite group G. We show that A is not characteristic in H in general, and further demonstrate in a series of examples that results, which hold about inner automorphisms of H, do not extend to the full automorphism group of H. We also give some new results about isoclinism and representation groups of G in the case that H is capable. Finally we give an example of two non-isomorphic groups of the same order, which not only have a representation group in common, but also have identical projective character tables.