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In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
If a set is such that it cannot be endowed with a group structure, then it is necessarily non-wellorderable. Otherwise the construction in the second section does yield a group structure. However these properties are not equivalent. Namely, it is possible for sets which cannot be well-ordered to have a group structure.
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
Download as PDF; Printable version; ... a group is a set, ... Group representation; Group structure and the axiom of choice;
Technically speaking, tagged PDF is a stylized use of the format that builds on the logical structure framework introduced in PDF 1.3. Tagged PDF defines a set of standard structure types and attributes that allow page content (text, graphics, and images) to be extracted and reused for other purposes. [31]
This abelian group is a Klein four-group-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element). In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M.
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field.
The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may ...