- 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/3Z 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/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 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 E6 as well, harv|Wilson|2009|p=2] At about the same time, it was shown that a family of five groups, called theMathieu 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
p-group , for instance.] 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.See also
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Semisimple group References
Notes
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