Socle (mathematics)In mathematics, the term socle has several related meanings. In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups.
ParisParis (ˈpærɪs; paʁi) is the capital and most populous city of France, with an official estimated population of 2,102,650 residents as of 1 January 2023 in an area of more than , making it the fourth-most populated city in the European Union as well as the 30th most densely populated city in the world in 2022. Since the 17th century, Paris has been one of the world's major centres of finance, diplomacy, commerce, culture, fashion, gastronomy and many areas.
Radical of a moduleIn mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(M) of M. Let R be a ring and M a left R-module. A submodule N of M is called maximal or cosimple if the quotient M/N is a simple module. The radical of the module M is the intersection of all maximal submodules of M, Equivalently, These definitions have direct dual analogues for soc(M).
CasablancaCasablanca, also known by its Arabic name Dar al-Bayda (ad-Dār al-Bayḍāʾ, adˈdaːru ɫbajdʕaːʔ, (): "White House"; Taddart Tumlilt), is the largest city in Morocco and the country's economic and business center. Located on the Atlantic coast of the Chaouia plain in the central-western part of Morocco, the city has a population of about 3.71 million in the urban area, and over 4.27 million in the Greater Casablanca, making it the most populous city in the Maghreb region, and the eighth-largest in the Arab world.
BrusselsBrussels (Bruxelles bʁysɛl or bʁyksɛl; Brussel ˈbrʏsəl), officially the Brussels-Capital Region (Région de Bruxelles-Capitale; Brussels Hoofdstedelijk Gewest), is a region of Belgium comprising 19 municipalities, including the City of Brussels, which is the capital of Belgium. The Brussels-Capital Region is located in the central portion of the country and is a part of both the French Community of Belgium and the Flemish Community, but is separate from the Flemish Region (within which it forms an enclave) and the Walloon Region.
Jules VerneJules Gabriel Verne (vɜrn; ʒyl gabʁijɛl vɛʁn; 8 February 1828 – 24 March 1905) was a French novelist, poet, and playwright. His collaboration with the publisher Pierre-Jules Hetzel led to the creation of the Voyages extraordinaires, a series of bestselling adventure novels including Journey to the Center of the Earth (1864), Twenty Thousand Leagues Under the Seas (1870), and Around the World in Eighty Days (1872). His novels, always well documented, are generally set in the second half of the 19th century, taking into account the technological advances of the time.
Conceptual artConceptual art, also referred to as conceptualism, is art in which the concept(s) or idea(s) involved in the work take precedence over traditional aesthetic, technical, and material concerns. Some works of conceptual art, sometimes called installations, may be constructed by anyone simply by following a set of written instructions.
