| Basic notions in group theory | ||||
| category of groups | ||||
|---|---|---|---|---|
| subgroups, normal subgroups | ||||
| quotient groups | ||||
| group homomorphisms, kernel, image | ||||
| (semi-)direct product, direct sum | ||||
| types of groups | ||||
| finite, infinite | ||||
| discrete, continuous | ||||
| multiplicative, additive | ||||
| abelian, cyclic, simple, solvable | ||||
In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a trivial group is a group consisting of a single element In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, a trivial group is a group consisting of a single element
For example, the cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French
One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group A5 of order 60, and every simple group of order 60 is isomorphic to A5. In Mathematics, an alternating group is the group of Even permutations of a Finite set. In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and it is possible to prove that every simple group of order 168 is isomorphic to PSL(2,7). In Mathematics, the Projective special linear group PSL(27 is a finite Simple group that has important applications in Algebra, In Mathematics, the Projective special linear group PSL(27 is a finite Simple group that has important applications in Algebra,
The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This is expressed by the Jordan-Hölder theorem. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.
The famous theorem of Feit and Thompson states that every group of odd order is solvable. In Mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. Walter Feit ( October 26[[ 930]] - July 29[[ 004]] was a Mathematician who worked in Finite group theory and Representation theory. John Griggs Thompson (born October 13 1932 in Ottawa Kansas, USA) is a Mathematician noted for his work in the field of Finite In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Therefore every finite simple group has even order unless it is cyclic of prime order.
Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these. In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups This page is about the infinite Thompson groups F T and V For the sporadic finite simple group Th see Thompson group (finite.
The Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. In finite group theory, the Schreier conjecture asserts that the group of Outer automorphisms of every finite simple group is solvable. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally This can be proved using the classification theorem.
See also
References
In mathematics the term semisimple is used in a number of related ways within different subjects PlanetMath is a free, collaborative online Mathematics Encyclopedia.Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic