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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Burnside's theorem in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
states that if ''G'' is a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
of
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
p^a q^b where ''p'' and ''q'' are
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s, and ''a'' and ''b'' are
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, then ''G'' is solvable. Hence each non-Abelian
finite simple group 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, or one of 26 sporadic groups. The list below gives all finite simple g ...
has order divisible by at least three distinct primes.


History

The theorem was proved by using the
representation theory of finite groups The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
. Several special cases of the theorem had previously been proved by Burnside in 1897,
Jordan Jordan, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian ter ...
in 1898, and
Frobenius Frobenius is a surname. Notable people with the surname include: * Ferdinand Georg Frobenius (1849–1917), mathematician ** Frobenius algebra ** Frobenius endomorphism ** Frobenius inner product ** Frobenius norm ** Frobenius method ** Frobenius g ...
in 1902. John G. Thompson pointed out that a proof avoiding the use of representation theory could be extracted from his work in the 1960s and 1970s on the N-group theorem, and this was done explicitly by for groups of odd order, and by for groups of even order. simplified the proofs.


Proof

The following proof — using more background than Burnside's — is by
contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. Let ''paqb'' be the smallest product of two prime powers, such that there is a non-solvable group ''G'' whose order is equal to this number. :*''G'' is a
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
with trivial center and ''a'' is not zero. If ''G'' had a
nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The n ...
proper
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
''H'', then (because of the minimality of ''G''), ''H'' and ''G''/''H'' would be solvable, so ''G'' as well, which would contradict our assumption. So ''G'' is simple. If ''a'' were zero, ''G'' would be a finite q-group, hence
nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...
, and therefore solvable. Similarly, ''G'' cannot be abelian, otherwise it would be solvable. As ''G'' is simple, its center must therefore be trivial. :* There is an element ''g'' of ''G'' which has ''qd'' conjugates, for some ''d'' > 0. By the first statement of
Sylow's theorem In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...
, ''G'' has a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
''S'' of order ''pa''. Because ''S'' is a nontrivial ''p''-group, its center ''Z''(''S'') is nontrivial. Fix a nontrivial element g\in Z(S). The number of conjugates of ''g'' is equal to the index of its
stabilizer subgroup In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under func ...
''Gg'', which divides the
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''qb'' of ''S'' (because ''S'' is a subgroup of ''Gg''). Hence this number is of the form ''qd''. Moreover, the integer ''d'' is strictly positive, since ''g'' is nontrivial and therefore not central in ''G''. :* There exists a nontrivial
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
ρ with character χ, such that its dimension ''n'' is not divisible by ''q'' and the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
''χ''(''g'') is not zero. Let (''χ''''i'')1 ≤ ''i'' ≤ ''h'' be the family of irreducible characters of ''G'' over \mathbb (here ''χ''1 denotes the trivial character). Because ''g'' is not in the same conjugacy class as 1, the orthogonality relation for the columns of the group's character table gives: : 0=\sum_^h \chi_i(1)\chi_i(g)= 1 + \sum_^h \chi_i(1)\chi_i(g). Now the ''χ''''i''(''g'') are
algebraic integer In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s, because they are sums of
roots of unity In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...
. If all the nontrivial irreducible characters which don't vanish at ''g'' take a value divisible by ''q'' at 1, we deduce that : -\frac1q=\sum_\fracq\chi_i(g) is an algebraic integer (since it is a sum of integer multiples of algebraic integers), which is absurd. This proves the statement. :* The complex number ''q''''d''''χ''(''g'')/''n'' is an algebraic integer. The set of integer-valued class functions on ''G'', ''Z''(\Z 'G'', is a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, finitely generated over \Z. All of its elements are thus integral over \Z, in particular the mapping ''u'' which takes the value 1 on the conjugacy class of g and 0 elsewhere. The mapping A\colon Z(\Z \rightarrow \operatorname(\Complex^n) which sends a class function ''f'' to : \sum_ f(s)\rho(s) is a ring homomorphism. Because \rho(s)^ A(u) \rho(s) = A(u) for all ''s'', Schur's lemma implies that A(u) is a
homothety In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point by the rule, : \o ...
\lambda I_n. Its
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album), by Nell Other uses in arts and entertainment * ...
''nλ'' is equal to : \sum_ f(s)\chi(s)=q^d\chi(g). Because the homothety ''λI''''n'' is the homomorphic image of an integral element, this proves that the complex number ''λ'' = ''q''''d''''χ''(''g'')/''n'' is an algebraic integer. :* The complex number ''χ''(''g'')/''n'' is an algebraic integer. Since ''q'' is relatively prime to ''n'', by
Bézout's identity In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Here the greatest common divisor of and is taken to be . The integers and are called B� ...
there are two integers ''x'' and ''y'' such that: : xq^d + yn=1\quad\text\quad \frac=x\frac + y\chi(g). Because a linear combination with integer coefficients of algebraic integers is again an algebraic integer, this proves the statement. :* The image of ''g'', under the representation ''ρ'', is a homothety. Let ''ζ'' be the complex number ''χ''(''g'')/''n''. It is an algebraic integer, so its norm ''N''(''ζ'') (i.e. the product of its conjugates, that is the roots of its minimal polynomial over \Q) is a nonzero integer. Now ''ζ'' is the average of roots of unity (the eigenvalues of ''ρ''(''g'')), hence so are its conjugates, so they all have an absolute value less than or equal to 1. Because the absolute value of their product ''N''(''ζ'') is greater than or equal to 1, their absolute value must all be 1, in particular ''ζ'', which means that the eigenvalues of ''ρ''(''g'') are all equal, so ''ρ''(''g'') is a homothety. :* Conclusion Let ''N'' be the kernel of ''ρ''. The homothety ''ρ''(''g'') is central in Im(''ρ'') (which is canonically isomorphic to ''G''/''N''), whereas ''g'' is not central in ''G''. Consequently, the normal subgroup ''N'' of the simple group ''G'' is nontrivial, hence it is equal to ''G'', which contradicts the fact that ''ρ'' is a nontrivial representation. This contradiction proves the theorem.


References

* * * *James, Gordon; and Liebeck, Martin (2001). ''Representations and Characters of Groups'' (2nd ed.)
Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
. . See chapter 31. *{{citation, mr=0323890 , last=Matsuyama, first= Hiroshi , title=Solvability of groups of order 2''a''''q''''b''. , journal=Osaka Journal of Mathematics, volume= 10 , year=1973, pages= 375–378 Theorems about finite groups