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A group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. class function A class function on a group G is a function that it is constant on the conjugacy classes of G. class number The class number of a group is the number of its conjugacy classes. commutator
If is such a group, and has order , then must be prime, since otherwise Cauchy's theorem applied to the (finite) subgroup generated by produces an element of order less than . Moreover, every finite subgroup of G {\displaystyle G} has order a power of p {\displaystyle p} (including G {\displaystyle G} itself, if it is finite).
The set of class functions of a group G with values in a field K form a K-vector space.If G is finite and the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by , = | | () ¯ where |G| denotes the order of G and bar is conjugation in the field K.
In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.
The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its k {\displaystyle k} -points are isomorphic as a group to the additive group of k {\displaystyle k} ), and usually denoted by G a {\displaystyle \mathrm {G} _{a}} .
It is known that a CLT group must be solvable and that every supersolvable group is a CLT group. However, there exist solvable groups that are not CLT (for example, A 4) and CLT groups that are not supersolvable (for example, S 4, the symmetric group of degree 4). There are partial converses to Lagrange's theorem.
The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term for programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class ...