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Encyclopedia > Simple group

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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This picture illustrates how the hours in a clock form a group. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element gâˆ’1ng is still in N. The statement N is a normal subgroup of G is written: . There are... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

For example, the cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. 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. 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. Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... The integers are commonly denoted by the above symbol. ...

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. 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. 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 group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... 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 a way that respects the given group operations. ... The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ... The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ...

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. This is expressed by the Jordan-Hölder theorem. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982. In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type (including the Tits group, which strictly speaking is not of Lie type), or one of 26 sporadic groups. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ... The integers are commonly denoted by the above symbol. ... In mathematics, a composition series of a group G is a normal series such that each Hi is a maximal normal subgroup of Hi+1. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore every finite simple group has even order unless it is cyclic of prime order. In mathematics, the Feit-Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. ... Walter Feit (October 26, 1930 - July 29, 2004) was a mathematician who worked in finite group theory and representation theory. ... John Griggs Thompson (born 13 Oct 1932) is a mathematician noted for his work in the field of finite groups. ... 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 realized by Galois theory. ...

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 Lie group which is also a simple group. ... Th see Thompson group (finite). ...

The Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. This can be proved using the classification theorem. In finite group theory, the Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. ... 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 realized by Galois theory. ... Results from FactBites:

 Simple group - Wikipedia, the free encyclopedia (208 words) 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. 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. Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these.
 Simple Lie group - Wikipedia, the free encyclopedia (498 words) These groups, and groups closely related to them, include many of the so-called classical groups of geometry, which lie behind projective geometry and other geometries derived from it by the Erlangen programme of Felix Klein. The complete list of simple Lie groups is the basis for the theory of the semisimple Lie groups and reductive groups, and their representation theory. Such groups are classified using the prior classification of the complex simple Lie algebras: for which see the page on root systems.
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