A translation of "Verallgemeinerung des Sylow'schen Satzes" by F. G. Frobenius
aa r X i v : . [ m a t h . HO ] A ug A translation of“Verallgemeinerung des Sylow’schen Satzes”by F. G. FrobeniusSitzungsberichte der Königl. Preuß. Akad. derWissenschaften zu Berlin, 1895 (II), 981–993. [ ] A generalization of S
YLOW ’s theorem.
By G. F
ROBENIUS .———
Every finite group whose order is divisible by the prime p contains elements oforder p . (C AUCHY , Mémoire sur les arrangements que l’on peut former avec des lettresdonnées . Exercises d’analyse et de physique Mathématique, Vol. III, §. XII, p. 250.)Their number is, as I will show here, always a number of the form ( p − )( np + ) .From that theorem, S YLOW deduced the more general one, that a group whoseorder h is divisible by p κ , must contain subgroups of order p κ . ( Théorèmes sur lesgroupes de substitutions , Math. Ann., Vol. V.) I gave a simple proof thereof in mywork
Neuer Beweis des S YLOW ’schen Satzes , C
RELLE ’s Journal, Vol. 100. The numberof those subgroups must, as I will show here, always be ≡ ( mod p ) . If p λ is thehighest power of p contained in h , then S YLOW proved this theorem only for thecase that κ = λ . Then any two groups of order p λ contained in H are conjugate,and their number np + h , while for κ < λ this does not hold ingeneral. I obtain the stated results in a new way from a theorem of group theorythat appears to be unnoticed thus far: In a group of order h, the number of elements whose order divides g is divisibleby the greatest common divisor of g and h. §. 1.If p is a prime number then any group P of order p λ has a series of invariantsubgroups (chief series) P , P , . . . , P λ − of orders p , p , . . . , p λ − , each of whichis contained in the subsequent one. S YLOW (loc. cit., p. 588) derives this resultfrom the theorem:I.
Every group of order p λ contains an invariant element of order p. An invariant element of a group H is an element of H that is permutable withevery element of H . If P contains the invariant element P of order p then thepowers of P form an invariant subgroup P of P whose order is p . Likewise, P / P has an invariant subgroup P / P of order p hence P has an invariantsubgroup P of order p which contains P , etc. In my work Über die Congruenznach einem aus zwei endlichen Gruppen gebildeten Doppelmodul , C
RELLE ’s Journal,Vol. 101 (§. 3, IV), I complemented that theorem with the following remark: F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] II.
Every group of order p λ − contained in a group of order p λ is an invariantsubgroup. Other proofs for this I developed in my work
Über endliche Gruppen , Sitzungs-berichte 1895 (§. 2, III, IV, V; §. 4, II). This can be obtained from Theorem I inthe following way: Let H be a group of order p λ , G a subgroup of order p λ − , P an invariant element of H whose order is p , and P the group of the powers of P .If G is divisible by P then G / P is an invariant subgroup of H / P because on canassume Theorem II as proven for groups whose order is smaller than p λ . Thus G isan invariant subgroup of H . If G is not divisible by P then H = GP , meaning thatevery element of H can be brought into the form H = G P , where G is an elementof G . Now, G is permutable with G and P even with every element of G . Hencealso H is permutable with G .The theorem mentioned at the onset lends itself to completion in a differentdirection:III. Every invariant subgroup of order p of a group of order p λ consists of powersof an invariant element. Let H be a group of order p λ , P an invariant subgroup of order p . If Q isany element of H and q = p κ is its order, then the powers of Q form a group Q contained in H of order q . If P is a divisor of Q then every element P of P is apower of Q , hence permutable with Q . If P is not a divisor of Q then P and Q arerelatively prime. P is permutable with every element of H and therefore with everyelement of Q . Thence PQ is a group of order p κ + and P is an invariant subgroupof it. But by Theorem II, Q is one also. Therefore P and Q are permutable in viewof the Theorem:IV. If each of the relatively prime groups A and B is permutable with everyelement of the other, then every element of A is permutable with every element of B . Indeed, if A is an element of A and B is an element of B , then the element A ( BA − B − ) = ( ABA − ) B − is contained in both A and B , and is therefore the principal element E .I want to prove Theorem III also in a second way: If Q − PQ = P a then Q − q PQ q = P a q . Hence if Q q = E then a q ≡ ( mod p ) . Now a p − ≡ ( mod p ) , hence as q and p − a ≡ ( mod p ) and therewith PQ = QP .Thirdly and finally, the Theorem follows from the more general Theorem:V. Every invariant subgroup of a group H of order p λ contains an invariantelement of H whose order is p. Partition the elements of H into classes of conjugate elements (conjugate withrespect to H ). If a class consists of a single element, then it is an invariant one, F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] and conversely every invariant element of H forms a class by itself. Let G be aninvariant subgroup of H and p κ its order. If the group G contains an element of aclass then it contains all its elements. Select an element G , G , . . . , G n from eachof the n classes contained in G . If the elements of H permutable with G ν forma group of order p λ ν , then the number of elements of H conjugate to G ν , i.e. thenumber of elements in the class represented by G ν , equals p λ − λ ν (C RELLE ’s Journal,Vol. 100, p. 181). Thence p κ = p λ − λ + p λ − λ + · · · + p λ − λ n .If G is the principal element E then λ = λ . Therefore not all the last n − p . There must existtherefore another index ν > λ ν = λ holds. Then G ν is an invariantelement of H whose order is p µ >
1, and the p µ − -th power of G ν is an invariantelement of H of order p that is contained in G .§. 2.I. If a and b are relative primes, then any element of order a b can always, andin a unique way, be written as a product of two elements whose orders are a and band which are permutable with each other. If A and B are two permutable elements whose orders a and b are relativeprimes, then AB = C has the order a b . Conversely, let C be any element of order a b . Determining the integer numbers x and y such that a x + b y = a x = β , b y = α , there holds C = C α C β , and C α has, since y is relatively prime to a , the order a , and C β the order b . (C AUCHY , loc. cit., §. V, p. 179.) Let now also C = AB , where A and B have the orders a and b and are permutable with eachother. Then C α = A α B α , B α = B b y = E , A α = A − β = A , thus A = C α and B = C β .Being powers of C , A and B belong to every group to which C belongs.II. If the order of a group is divisible by n then the number of those elements ofthe group whose order divides n is a multiple of n.
Let H be a group of order h and n a divisor of h . For every group whose order is h ′ < h and for each divisor n ′ of h ′ , I assume the Theorem as proven. The numberof elements of H whose order divides n is, if n = h holds, equal to n . So if n < h , Ican assume the theorem has been proven for every divisor of h which is > n . Nowif p is a prime dividing hn , then the number of elements of h whose order divides np is divisible by np , hence also by n . Let np = p λ r , where r is not divisible by p and λ ≥
1. Let K be the complex of those elements of H whose order divides np F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] but not n , hence divisible by p λ , and let k be the order of this complex. Then itonly remains to show that the number k , if it differs from zero, is divisible by n .For that purpose I prove that k is divisible by p λ − and r .I partition the elements of K into systems by assigning two elements to thesame system if each is a power of the other. All elements of a system have thesame order m . Their number is φ ( m ) . A system is completely determined byeach of its elements A , it is formed by the elements A µ where µ runs through the φ ( m ) numbers which are < m and relatively prime to m . If A is an element ofthe complex K then all the elements of the system represented by A belong to thecomplex K . Then the order m of A is divisible by p λ , hence also φ ( m ) by p λ − .Since the number of elements of each system, into which K is decomposed, isdivisible by p λ − , so must k be divisible by p λ − .To show secondly that k is also divisible by r , I partition again the elementsof K into systems, but of a different kind, yet still such that the cardinality ofelements of each system is divisible by r . Every element of K can, and in a uniqueway at that, be represented as a product of an element P of order p λ and a withit permutable element Q whose order divides r . Conversely, every product PQ soobtained belongs to the complex K .Let P be some element of order p λ . All elements of K that are permutable with P form a group Q whose order q is divisible by p λ . The powers of P form a group P of order p λ which is an invariant subgroup of Q . The elements Q of Q thatsatisfy the equation Y r = E are identical to those that satisfy the equation Y t = E ,where t is the greatest common divisor of q and r . The first issue is to determinethe number of those elements.Every element of Q can, and in a unique way at that, be represented as aproduct of an element A whose order is a power of p and a with it permutableelement B whose order is not divisible by p .If the t -th power of AB belongs to the group P then ( AB ) t = A t B t = P s , hence A t = P s , B t = E ,because also this element can be decomposed in the given fashion in a single way.Thus A t belongs to P , hence also A itself because t is not divisible by p . The orderof the group Q / P is qp λ < h . The number of (complex) elements of this group thatsatisfy the equation Y t = R is therefore a multiple of t , say tu . If P AB is suchan element then, as A belongs to P , P A = P , hence P AB = P B . Since B , as anelement of Q , is permutable with P , the complex P B contains only one elementwhose order divides t , namely B itself, whilst the order of every other element of F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] P B is divisible by p . Let PB + P B + P B + · · · be the tu distinct (complex) elements of the group Q / P whose t -th power iscontained in P , then this complex contains all those elements of Q whose t -thpower (absolutely) equals E . However, only the elements B , B , B , · · · have thisproperty. Thus Q contains exactly tu elements that satisfy the equation Y t = E ,or there are, if P is a certain element of order p λ , exactly tu elements that arepermutable with P and whose order divides r .The number of elements of H permutable with P is q . The number of elements P , P , P , · · · of H that are conjugate to P with respect to H is therefore hq . Thenthere are exactly tu elements Q in H that are permutable with P and whoseorder divides r . Taking each of the hq elements P , P , P , · · · successively as X andeach time as Y the tu elements permutable with X and satisfy the equation Y r = E ,one obtains the system K ′ of k ′ = hq tu distinct elements X Y of the complex K . Now h is divisible by both q and r hencealso by their least common multiple qrt . Thus k ′ is divisible by r . The system K ′ is completely determined by each of its elements. Two distinct systems among K ′ , K ′′ , · · · have no element in common. Their orders k ′ , k ′′ , · · · are all divisible by r . Thus also k = k ′ + k ′′ + · · · is divisible by r .The number of elements of a group that satisfy the equation X n = E is mn , theinteger number m is > X = E always satisfies that equation.III. If the order of a group H is divisible by n then the elements of H whose orderdivides n generate a characteristic subgroup of H whose order is divisible by n. Let R be the complex of elements of H that satisfy the equation X n = E . If X isan element of R and R is any element ∗ permutable with H then R − X R is also anelement of R . Thus R − R R = R . Let the complex R generate a group G of order g . Then also R − G R = G , so that G is a characteristic subgroup of H .If q µ is the highest power of a prime q that divides n then q µ also divides h .Thus H contains a group Q of order q µ . Now R is divisible by Q , hence also G , andconsequently g is divisible by q µ . Since this holds for every prime q that divides n , g is divisible by n .On the relation of the complex R to the group G I further note the following:I considered in
Über endliche Gruppen , §. 1 the powers R , R , R , · · · of a complex ∗ tn: cf. Frobenius, Über endliche Gruppen , §. 5, SB. Akad. Berlin, 1895 (I), http: // dx.doi.org / / e-rara-18846 F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] R . If in that sequence R r + s is the first one that equals one of the foregoing ones R r , then R ρ = R σ if and only if ρ ≡ σ ( mod s ) and ρ and σ are both ≥ r . Let t be the number uniquely defined by the conditions t ≡ ( mod s ) and r ≤ t < r + s .Then R t is the only group contained in that sequence of powers. If R contains theprincipal element E then R ρ + is divisible by R ρ . Hence G = R t is divisible by R . If N is an element of the group G then G N = G . More generally then, if R is a complex of elements contained in G then GR = G . Therefore R t + = R t ,hence s = t = r . Consequently, R r = R r + is the first one in the sequenceof powers of R that equals the subsequent one, and this is the group generated bythe complex R .IV. If the order of a group H is divisible by the two relatively prime numbers rand s, if there exists in H exactly r elements A whose order divides r and exactly selements B whose order divides s, then each of the r elements A is permutable witheach of the s elements B and there exist in H exactly rs elements whose order dividesrs, namely the rs distinct elements AB = BA.
Indeed, every element C of H whose order divides rs can be written as a prod-uct of two with each other permutable elements A and B whose orders divide r and s . Now H contains no more than r elements A and no more than s elements B . Were it not the case that each of the r elements A is permutable with each ofthe s elements B and furthermore that the rs elements AB are all distinct, then H would contain less than rs elements C . But this contradicts Theorem II.§. 3.If the order h of a group H divisible by the prime p then H contains elements oforder p , namely mp − mp elements in H whose orderdivides p . From this theorem of C AUCHY , S
YLOW derived the more general one,that any group whose order is divisible by p κ possesses a subgroup of order p κ . Inhis proof he draws on the language of the theory of substitutions. If one wants toavoid this, one should apply the procedure that I used in my work Über endlicheGruppen in the proof of Theorems V and VII, §. 2.Another proof is obtained by partitioning the mp − P of order p contained in H into classes of conjugate elements. If the elements of H permutablewith P form a group G of order g , then the number of elements conjugate to P is hg . Thus mp − = X hg F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] where the sum is to be extended over the different classes into which the elements P are segregated. From this equation it follows that not all the summands hg aredivisible by p . Let p λ be the highest power of p contained in h , and let κ ≤ λ .If hg is not divisible by p then g is divisible by p λ . The powers of P form a group P of order p , which is an invariant subgroup of G . The order of the group G / P is gp < h . For this group we may therefore assume the theorems which we wishto prove for H as known. Thus it contains a group P κ / P of order p κ − , andin the case that κ < λ , a group P κ + / P of order p κ that is divisible by P κ / P .Consequently, H contains the group P κ of order p κ and the group P κ + of order p κ + that is divisible by P κ . §. 4.I. If the order of a group is divisible by the κ -th power of the prime p then thenumber of groups of order p κ contained therein is a number of the form np + . Let r κ denote the number of groups of order p κ contained in H . Then thenumber of elements of H whose order is p equals r ( p − ) . As shown above, thisnumber has the form mp −
1. Thus r ≡ ( mod p ) . (1.)Let r κ − = r , r κ = s , and let A , A , · · · , A r (2.)be the r groups of order p κ − contained in H and B , B , · · · , B s (3.)the s groups of order p κ . Suppose the group A ρ is contained in a ρ of the groups(3.). Suppose the group B σ is divisible by b σ of the groups (2.). Then a + a + · · · + a r = b + b + · · · + b s (4.)is the number of distinct pairs of groups A ρ , B σ for which A ρ is contained in B σ .Let A be one of the groups (2.). Of the groups (3.) let B , B , · · · , B a be thosethat are divisible by A . By §. 3, a >
0, and by Theorem II, §. 1, A is an invariantsubgroup of each of these a groups, hence also of their least common multiple G .Therefore the group G / A contains the a groups B / A , B / A , · · · , B a / A of order F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] p and none further. Indeed, if B / A is a group of order p contained in G / A then B is a group of order p κ divisible by A . By formula (1.) there holds a ≡ ( mod p ) .Thus a ρ ≡ a + a + · · · + a r ≡ r ( mod p ) . (5.)Now I need the Lemma: The number of groups of order p λ − which are contained in a group of order p λ is ≡ ( mod p ) . I suppose this Lemma is already proven for groups of order p κ if κ < λ . Then,if in the above expansion κ < λ then b σ ≡ b + b + · · · + b s ≡ s ( mod p ) . (6.)Therefore r ≡ s or r κ − ≡ r κ ( mod p ) , and since this congruence holds for eachvalue κ < λ , it is 1 ≡ r ≡ r ≡ · · · ≡ r λ − ( mod p ) .Applying this result to a group H whose order is p λ , it is therefore r λ − ≡ ( mod p ) for such a group, and with this, the above Lemma is proven also forgroups of order p λ , if it holds for groups of order p κ < p λ , it is therefore generallyvalid. For each value κ consequently, r κ ≡ r κ − and therefore r κ ≡ ( mod p ) .In exactly the same way one proves the more general Theorem:II. If the order of a group H is divisible by the κ -th power of the prime p, if ϑ ≤ κ and P is a group of order p ϑ contained in H , then the number of groups of order p κ contained in H that are divisible by P is a number of the form np + . §. 5.The Lemma used in §. 4 can also be proven in the following way by relying onthe Theorem: Every group H of order p λ has a subgroup A of order p λ − and sucha subgroup is always an invariant one. Let A and B be two distinct subgroups oforder p λ − contained in H and let D be their greatest common divisor. Since A and B are invariant subgroups of H , so is D , and since H is the least common multipleof A and B , D has order p λ − . Thus H / D is a group of order p . Any such grouphas, depending on whether it is a cyclic group or not, 1 or p + p , thus in our case p +
1, since A / D and B / D are two distinct groups of this type.Therefore H contains exactly p + p λ − that are divisibleby D . F ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] The group H always contains a group A of order p λ − . If it contains yet anotherone, then H has an invariant subgroup D of order p λ − which is contained in A and for which the group H / D is not a cyclic one. Let D , D , · · · , D n be all thegroups of this kind. Then there exist in H besides A other p groups of order p λ − divisible by D A , A , · · · , A p , (1.)and likewise p groups that are divisible by D A p + , A p + , · · · , A p , (2.)etc., and finally p groups divisible by D n A ( n − ) p + , A ( n − ) p + , · · · , A np + . (3.)The np + A , A , · · · , A np are all the groups of order p λ − contained in H since each such group B has to have in common with A a certain divisor D whichis one of the n groups D , D , · · · , D n . They are, furthermore, all distinct. Indeed,if A = A p + was true then A would be divisible by both groups D and D , hencealso by their least common multiple A . If P is a group of order p ϑ contained in H then one can subject all the groups considered above to the condition of beingdivisible by P . If conversely H is an invariant subgroup of a group P of order p ϑ then one can require that they all be invariant subgroups of P .With the help of Theorem V, §.1 it is easy to prove that the number of groupsof order p λ − that are contained in a group of order p λ equals 1 only if H is a cyclicgroup.I. The number of invariant subgroups of order p κ contained in a group of orderp λ is a number of the form np + . Let H be a group of order h , let p λ be the highest power of p contained in h , let κ ≤ λ and P κ any group of order p κ contained in H . Each group P κ is containedin np + P κ into two kinds.For a group of the first kind there exists a group P λ of which P κ is an invariantsubgroup, for a group of the second kind no such group exists. The number ofelements of H permutable with P κ is divisible by p λ in the first case, and in thesecond case it is not. The number of groups conjugate to P κ is therefore divisibleby p in the second case, in the first case it is not. Hence diving the groups P κ intoclasses of conjugate groups one recognizes that the number of groups P κ of thesecond kind is divisible by p . Consequently, the number of groups P κ of the firstkind is ≡ ( mod p ) . ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] II. If H is a group of order p λ and G is an invariant subgroup of H whose orderis divisible by p κ then the number of groups of order p κ contained in G that areinvariant subgroups of H is a number of the form np + . Also here let more generally p λ be the highest power of the prime p that dividesthe order h of H . Let G be an invariant subgroup of H whose order g is divisibleby p κ . The number of all groups P κ of order p κ contained in G is ≡ ( mod p ) . Idivide them into groups of the first and the second kind (with respect to H ) andfurther into classes of conjugate groups. If G is divisible by P κ then G is alsodivisible by every group conjugate to P κ . Therefrom the claim follows in the sameway as above. One can also easily prove it directly by means of the method usedin §. 4:Let the order of H be h = p λ . By Theorem V, §.1 the group G contains elementsof order p that are invariant elements of H . They form, together with the principalelement, a group. If p α is its order then p α − H whose order is p consists of thepowers of such an element. Therefore there exist in G r = p α − p − groups of order p that are invariant subgroups of H . This number is r ≡ ( mod p ) . (4.)Let A , A , · · · , A r (5.)be those r groups and let B , B , · · · , B s (6.)be the s groups of order p κ contained in G that are invariant subgroups of H .Let B be one of the groups (6.). Among the groups (5.) let A , A , · · · , A b bethose contained in B . By (4.) is then b ≡ ( mod p ) . Let A be one of the groups(5.). Among the groups (6.) let B , B , · · · , B a be those divisible by A . Then B / A , B / A , · · · , B a / A are the groups of order p κ − contained in G / A that are in-variant subgroups of H / A . By the method of induction is therefore a ≡ ( mod p ) .Resorting to the same notation as in §. 4 there holds1 ≡ r ≡ a + a + · · · + a r ≡ b + b + · · · + b s ≡ s ( mod p ) .I add a few remarks on the number of groups P κ of the first kind that areconjugate to a particular one, and on the number of classes of conjugate groupsinto which the groups P κ are partitioned. ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] Let P be a group of order p λ contained in P and Q an invariant subgroup of P of order p κ . The elements of H permutable with P ( Q ) form a group of P ′ ( Q ′ ) of order p ′ ( q ′ ) . Let the greatest common divisor of P ′ and Q ′ be the group R of order r . The groups P ′ , Q ′ and R are divisible by P . Let p δ be the order ofthe largest common divisor of P and a group conjugate with respect to H that isselected in such a way that δ is a maximum. Then ( Über endliche Gruppen , §. 2,VIII) hp ′ ≡ ( mod p λ − δ ) .The group R consists of all the elements of Q ′ that are permutable with P . Withthis, q ′ r ≡ ( mod p λ − δ ) .Consequently, hq ′ ≡ p ′ r ( mod p λ − δ ) . (7.)Herein, hq ′ is the number of groups that are conjugate to Q with respect to H and p ′ r is the number of groups that are conjugate to Q with respect to P ′ . Indeed,the group R consists of all the elements of P ′ that are permutable with Q . Thenumber of groups in a certain class in H is therefore congruent ( mod p λ − δ ) to thenumber of groups in the corresponding class in P ′ .Furthermore, the number of distinct classes in H (into which the groups P κ of the first kind are partitioned) equals the number of those classes in P ′ . Thisfollows from the Theorem:III. If two invariant subgroups of P are conjugate with respect to H then so theyare with respect to P ′ . Let Q and Q be two invariant subgroups of P . If they are conjugate withrespect to H then there exists in H such an element H that H − Q H = Q (4.)holds. Since Q is an invariant subgroup of P , H − Q H = Q is an invariantsubgroup of H − P H = P . ROBENIUS : A generalization of S
YLOW ’s theorem. [ ] Hence Q ′ is divisible by P and P . Consequently ( Über endliche Gruppen , §. 2,VII) there exists in Q ′ such an element Q that Q − P Q = P ,hence P HQ = HQ P holds. Thus HQ = P is an element of P ′ . Inserting the expression H = PQ − intothe equation (4.) one obtains, since Q is permutable with Q , P − Q P = Q − Q Q = Q .There exists therefore in P ′ an element P that transforms Q into Q .Partition now the groups P κ contained in H (of the first kind) into classes ofconjugate groups (with respect to H ) and choose from each class a representative.If Q is one, then Q is a group of order p κ which is contained in a certain group P as an invariant subgroup. If H − P H = P then H − Q H = Q is an invariantsubgroup of P . One can therefore choose the representatives of different classes insuch a way that they are all invariant subgroups of a certain group P of order p λ .Each invariant subgroup of P of order p κ is then conjugate to one of these groupswith respect to H , hence also with respect to P ′ . Let the invariant subgroups P κ of P aggregate into s classes of groups that are conjugate with respect to P ′ . Thenthe groups P κ of the first kind of H also aggregate into s classes of groups that areconjugate with respect to H . Translated with minor typographical corrections fromF. G. Frobenius, “Verallgemeinerung des Sylow’schen Satzes”, Sitzungsberichte derKönigl. Preuß. Akad. der Wissenschaften zu Berlin, 1895 (II), 981–993.http: // dx.doi.org / / e-rara-18880For the terminology seeFrobenius, Über endliche Gruppen , SB. Akad. Berlin, 1895 (I), 163–194.http: // dx.doi.org / //