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Examples of transformations with different orders: 90° rotation with order 4, shearing with infinite order, and their compositions with order 3. In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite.
order of a group The order of a group (G, •) is the cardinality (i.e. number of elements) of G. A group with finite order is called a finite group. order of a group element The order of an element g of a group G is the smallest positive integer n such that g n = e. If no such integer exists, then the order of g is said to be infinite.
For example, Poole (see below) found in his empirical research that seemingly complex patterns of behavior in group decision-making result from the interplay of life-cycle and teleological motors. An important observation made by McGrath and Tschan in 2004 regarding the different models of group development found in the literature is that ...
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
Cauchy's theorem is generalized by Sylow's first theorem, which implies that if p n is the maximal power of p dividing the order of G, then G has a subgroup of order p n (and using the fact that a p-group is solvable, one can show that G has subgroups of order p r for any r less than or equal to n).
Groupthink is sometimes stated to occur (more broadly) within natural groups within the community, for example to explain the lifelong different mindsets of those with differing political views (such as "conservatism" and "liberalism" in the U.S. political context [7] or the purported benefits of team work vs. work conducted in solitude). [8]
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
Group dynamics is a system of behaviors and psychological processes occurring within a social group (intragroup dynamics), or between social groups (intergroup dynamics). The study of group dynamics can be useful in understanding decision-making behaviour, tracking the spread of diseases in society, creating effective therapy techniques, and ...