- Simple group
In

mathematics , a**simple group**is a group which is not thetrivial group and whose onlynormal subgroup s are thetrivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and thequotient group , and the process can be repeated. If the group is finite, then eventually one arrives at uniquely determined simple groups by theJordan-Hölder theorem .**Examples**For example, the

cyclic group "G" =**Z**/3**Z**ofcongruence class es modulo 3 (seemodular arithmetic ) is simple. If "H" is a subgroup of this group, its order (the number of elements) must be adivisor 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**/12**Z**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 anabelian group is normal. Similarly, the additive group**Z**ofinteger s is not simple; the set of even integers is a non-trivial proper normal subgroup.One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the

cyclic group s of prime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is thealternating group "A"_{5}of order 60, and every simple group of order 60 is isomorphic to "A"_{5}. The second smallest nonabelian simple group is the projective special linear groupPSL(2,7) of order 168, and it is possible to prove that every simple group of order 168 is isomorphic toPSL(2,7) .**Classification**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 number s are the basic building blocks of theinteger s. This is expressed by the . In a huge collaborative effort, theclassification of finite simple groups was accomplished in 1982.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.

Simple groups of infinite order also exist:

simple Lie group s and the infiniteThompson groups "T" and "V" are examples of these.The

Schreier conjecture asserts that the group ofouter automorphism s of every finite simple group is solvable. This can be proved using the classification theorem.**poradic simple groups**In 1831

Évariste Galois discovered that thealternating group s on five or more points were simple. The next discoveries were byCamille Jordan in 1870. Jordan had found 4 families of simple matrix groups overfinite field s of prime order. Later Jordan's results were generalized to arbitrary finite fields byLeonard Dickson . In the process he discovered several new infinite families of groups, now called theclassical group s. [*Actually, Dickson also constructed groups of type G2 and E*] At about the same time, it was shown that a family of five groups, called the_{6}as well, harv|Wilson|2009|p=2Mathieu group s and first described byÉmile Léonard Mathieu in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "sporadic" byWilliam Burnside in his 1897 textbook. In 1981Robert Griess announced that he had constructedBernd Fischer 's "Monster group ". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Each element of the Monster can be expressed as a 196,883 by 196,883 matrix. Soon after a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully listed all finite simple groups.**ylow test for nonsimplicity**Let "n" be a positive integer that is not prime, and let "p" be a prime divisor of "n". If 1 is the only divisor of "n" that is equal to 1 modulo p, then there does not exist a simple group of order "n".

Proof: If "n" is a prime-power, then a group of order "n" has a nontrivial center [

*See the proof in*] and, therefore, is not simple. If "n" is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order "n" is equal to 1 modulo "p" and divides "n". Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.p-group , for instance.**See also***

Semisimple group **References****Notes****Textbooks***

***External links***

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**SIMPLE Group**— Infobox Company company name = SIMPLE Group company company type = Private (Limited liability) foundation = flagicon|GIB Gibraltar (Incorporation: 1999) location city = Gibraltar location country = Gibraltar key people = Fabien Bénédicte Suant,… … Wikipedia**simple group**— Math. a group that has no normal subgroup except the group itself and the identity. * * * … Universalium**simple group**— Math. a group that has no normal subgroup except the group itself and the identity … Useful english dictionary**Characteristically simple group**— In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups.… … Wikipedia**Absolutely simple group**— In mathematics, in the field of group theory, a group is said to be absolutely simple if it has no proper nontrivial serial subgroups. That is, G is an absolutely simple group if the only serial subgroups of G are { e } (the trivial subgroup),… … Wikipedia**Strictly simple group**— In mathematics, in the field of group theory, a group is said to be strictly simple if it has no proper nontrivial ascendant subgroups. That is, G is a strictly simple group if the only ascendant subgroups of G are { e } (the trivial subgroup),… … Wikipedia**Group theory**— is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… … Wikipedia**Simple Lie group**— Lie groups … Wikipedia**Group (mathematics)**— This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia**Group of Lie type**— In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… … Wikipedia