Burnside theorem

The set of Burnside is a set of the mathematical branch of group theory and states that groups a certain order automatically resolved are.

formulation

A finite group of order

{\ displaystyle p ^ {a} q ^ {b}}

,

where and are prime numbers and and integers greater than or equal to 0 can be resolved. ${\ displaystyle p}$ ${\ displaystyle q}$ ${\ displaystyle a}$ ${\ displaystyle b}$

Hence every nonabelian finite simple group has an order divisible by at least three different prime numbers .

history

The theorem was proved in 1904 by William Burnside using group representation theory . Some partial results had previously been achieved by Burnside, Jordan and Frobenius . Thompson then pointed out a possible line of evidence that avoids the use of representation theory and could be gleaned from his work on N groups. This was then carried out by Goldschmidt for groups of odd order and by Bender for groups of even order. Matsuyama further simplified this evidence.

Sketch of the Burnside Evidence

The following sketch of evidence must remain incomplete in this brief, but gives an impression of the methods used. A German-language elaboration of this proof can be found in Meyberg's textbook given below .

If an irreducible complex character of a finite group is then all over , where be an element of the conjugation class . ${\ displaystyle \ chi}$ ${\ displaystyle G}$ ${\ displaystyle | K | \ chi (k) / \ chi (1)}$ ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle k}$ ${\ displaystyle K}$

If and are coprime , one shows using 1) that either is 0 or has the absolute value . ${\ displaystyle | K |}$ ${\ displaystyle \ chi (1)}$ ${\ displaystyle \ chi (k)}$ ${\ displaystyle \ chi (1)}$

By means of 2) it follows that a finite group cannot be simple if it has a non-trivial conjugation class of power . According to the orthogonality relations , there must be a non-trivial character with a degree that is too coprime, whose value is too coprime. According to 2) has the absolute amount , which has the consequence that the corresponding irreducible representation is mapped to a multiple of the identical operator . The irreducible representation is thus non-trivial and one-dimensional, which is why the kernel is a non-trivial normal divisor. ${\ displaystyle K}$ ${\ displaystyle p ^ {n}}$ ${\ displaystyle | K |}$ ${\ displaystyle \ chi (k)}$ ${\ displaystyle p}$ ${\ displaystyle \ chi (k)}$ ${\ displaystyle \ chi (1)}$ ${\ displaystyle k}$ ${\ displaystyle \ chi}$

The class equation now shows that a group of the order has a non-trivial to coprime conjugation class, which must then be of the size for a . After the previous step it follows that it cannot be easy.

{\ displaystyle p ^ {a} q ^ {b}, \, a, b> 0}

{\ displaystyle q}

{\ displaystyle p ^ {r}}

{\ displaystyle r> 0}

{\ displaystyle G}

Induction on the group order finally shows that every group of such order has non-trivial normal divisors and therefore the group must be solvable.

Smallest non-resolvable group

According to a further, simpler theorem, every group whose order is the product of three prime numbers can also be resolved. Together with Burnside's theorem, the order of a non-resolvable group must therefore be the product of at least four prime numbers, three of which are different from one another, that is, the smallest possible order . The group A ₅ shows that there is indeed a non-resolvable group of order 60. ${\ displaystyle 2 ^ {2} \ times 3 \ times 5 = 60}$

Individual evidence

^ W. Burnside: On Groups of Order p ^α q ^β , Proc. London Math. Soc. (1904), pp. 388-392.
^ DM Goldschmidt: A group theoretic proof of the p ^a q ^b theorem for odd primes , Math. Z. (1970), Volume 11, pages 373-375
^ H. Bender: A group theoretic proof of Burnside's p ^a q ^b -theorem. , Math. Z. (1972), Volume 126, pages 327-338
↑ H. Matsuyama: Solvability of groups of order 2 ^a q ^b , Osaka J. Math. (1973), Volume 10, pages 375-378
^ K. Meyberg: Algebra Part 2 , Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , sentence 9.7.2: p ^a p ^b sentence by Burnside
↑ B. Huppert: Finite Groups I , Springer-Verlag (1967), chap. I, §8, sentence 8.13

[1] W. Burnside: On Groups of Order p ^α q ^β , Proc. London Math. Soc. (1904), pp. 388-392.

[2] DM Goldschmidt: A group theoretic proof of the p ^a q ^b theorem for odd primes , Math. Z. (1970), Volume 11, pages 373-375

[3] H. Bender: A group theoretic proof of Burnside's p ^a q ^b -theorem. , Math. Z. (1972), Volume 126, pages 327-338

[4] H. Matsuyama: Solvability of groups of order 2 ^a q ^b , Osaka J. Math. (1973), Volume 10, pages 375-378

[5] K. Meyberg: Algebra Part 2 , Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , sentence 9.7.2: p ^a p ^b sentence by Burnside

[6] B. Huppert: Finite Groups I , Springer-Verlag (1967), chap. I, §8, sentence 8.13