Systems Of Imprimitivity Research Articles

Integral systems of imprimitivity

Integral systems of imprimitivity

Positive definite measures

In this paper we prove two theorems relating positive definite measures to induced representations. The first shows how the injection of a positive definite measure on a topological group H into a containing locally compact group G in which H is closed gives rise to induced representations. The second is another version of Mackey's imprimitivity theorem, along the lines of Loomis' proof [5]. We feel this is justified on several grounds. Firstly, our proof is simpler than Loomis'. We make no use of the Radon-Nikodym theorem nor of quasi-invariant measures. Secondly, we do not assume in advance that our system of imprimitivity is based on the reduced algebra of Borel sets in G/H. Instead, this fact is seen as a consequence of the theorem. Finally, the statement and proof of Theorem 2 in [5] are in need of minor repairs. Using Loomis' notation, the induced representation space of Vis spanned, not by the set of functions {fu: uCH}, but rather by the set { [E]fu: uCH, E a Borel subset of G/K}. Formula (8) must then be replaced by formula (11) in the statement of the theorem. The algebra Co(SXG) used in the present paper may be looked upon as a device for accomplishing these changes. All nonobvious definitions, notations, and conventions are those of [1i].

A remark on systems of imprimitivity

A remark on systems of imprimitivity

Representations Induced in an Invariant Subgroup

The object of this paper is to investigate the representation 21H induced by an irreducible representation 2t1 of a group G in an invariant subgroup H of G. In ?1 it is shown that 21, is either itself irreducible or is fully reducible into conjugate irreducible representations of H. In ?2 it is shown that 2IG is imprimitive unless all the irreducible components of 21H are equivalent. In fact, if T is the representation space of Kg, and hence also of 21H , and if we lump together all equivalent subspaces of T under WH , then the resulting subspaces T1 , T2 , T. , * constitute a system of imprimitivity of WsG. If we define G' to be the subgroup of G leaving one of these invariant, say T, then the component of 21G' in T, is an irreducible representation 2<;, of G', and 21G is expressible very simply in terms of WG, . These results hold for any group G and any ground-field P. In ??3-5, however, we make the assumption that P is algebraically closed. In ?3 it is found that [' , is the direct product of two irreducible projective representations of G', one of which is actually a projective representation r of the factor-group G'/H. In ?4 some progress is made on the question of whether or not a given irreducible representation of H can be embedded in some irreducible representation of G, and in ?5 we consider all possible ways of doing this. Two irreducible representations 21G and TG of G are said to be associate if 21H and EH have an irreducible component in common; associates differ only in the projective representation r of G'/H mentioned above. In the case when the factor-group G/H is a finite cyclic group of order k, associates can be described as differing from each other only by a factor which is a one-dimensional representation of G/H, and hence just a kth root of unity. In the simplest case of all, when H is of index two in G, WG has just one associate 2* (besides itself) differing from 2tI only in that we change the sign of the matrices corresponding to elements of G not in H. The situation may then be described as follows. If 21G is not equivalent to * , then WH is irreducible. If t is equivalent to * , then 96f decomposes into two inequivalent (conjugate) irreducible components. If TG is another irreducible representation of G equivalent to neither Ads nor 2*, then TH can have no irreducible component in common with 2I(= 2H)Virtually all of this theory is known in the case of a finite group G, and the greater part of it goes back to Frobenius. For the decomposition of WH into conjugates we must refer to Frobenius' original paper.' For the results of ?2-

Representations Induced in an Invariant Subgroup

The object of this paper is to investigate the representation 21H induced by an irreducible representation 2t1 of a group G in an invariant subgroup H of G. In ?1 it is shown that 21, is either itself irreducible or is fully reducible into conjugate irreducible representations of H. In ?2 it is shown that 2IG is imprimitive unless all the irreducible components of 21H are equivalent. In fact, if T is the representation space of Kg, and hence also of 21H , and if we lump together all equivalent subspaces of T under WH , then the resulting subspaces T1 , T2 , T. , * constitute a system of imprimitivity of WsG. If we define G' to be the subgroup of G leaving one of these invariant, say T, then the component of 21G' in T, is an irreducible representation 2<;, of G', and 21G is expressible very simply in terms of WG, . These results hold for any group G and any ground-field P. In ??3-5, however, we make the assumption that P is algebraically closed. In ?3 it is found that [' , is the direct product of two irreducible projective representations of G', one of which is actually a projective representation r of the factor-group G'/H. In ?4 some progress is made on the question of whether or not a given irreducible representation of H can be embedded in some irreducible representation of G, and in ?5 we consider all possible ways of doing this. Two irreducible representations 21G and TG of G are said to be associate if 21H and EH have an irreducible component in common; associates differ only in the projective representation r of G'/H mentioned above. In the case when the factor-group G/H is a finite cyclic group of order k, associates can be described as differing from each other only by a factor which is a one-dimensional representation of G/H, and hence just a kth root of unity. In the simplest case of all, when H is of index two in G, WG has just one associate 2* (besides itself) differing from 2tI only in that we change the sign of the matrices corresponding to elements of G not in H. The situation may then be described as follows. If 21G is not equivalent to * , then WH is irreducible. If t is equivalent to * , then 96f decomposes into two inequivalent (conjugate) irreducible components. If TG is another irreducible representation of G equivalent to neither Ads nor 2*, then TH can have no irreducible component in common with 2I(= 2H)Virtually all of this theory is known in the case of a finite group G, and the greater part of it goes back to Frobenius. For the decomposition of WH into conjugates we must refer to Frobenius' original paper.' For the results of ?2-

Number of Systems of Imprimitivity of Transitive Substitution Groups

Number of Systems of Imprimitivity of Transitive Substitution Groups

The subgroup composed of the substitutions which omit a letter of a transitive group

In a transitive group of order g and of degree n the subgroup G, composed of all the substitutions of G which omit a given letter is of order gin. The properties of G, frequently throw light on the possible properties of G. In particular, when G1 is transitive and of degree n-1, G must be multiply transitive, but when G, is of a lower degree than n-1, there must be at least one substitution besides identity which is commutative with every substitution of G. If the order of such a substitution is less thani n its systems of intransitivity are systems of imprimitivity of G. Hence it results that, when G is primitive and G, is not of degree n-1, G must be the regular group of prime order p. While a number of such properties relating to G, have been known for a long time little has been done towards determining the possible substitution groups which involve a particular G,. For instance, when G, is the symmetric group of degree mn, G must be the symmetric group of degree m 4+1, except in the special case when m 2. In this case it may also be the octic group of degree 4, as is well known, but it cannot be any other group. This theorem results directly from the facts that there is no substitution on the letters of the symmetric group of degree mn> 2 which is commutative with every substitution of this symmetric group and that a regular group of degree m can always be used as the G, for at least one transitive group of degree 2in and of order 2 mn. In a similar manner it results that every alternating group, except the alternating group of degree 3, can appear as the G, of only the alternating group of the next larger degree, while the alternating group of degree 3 is also the G, of a group of order 18 and of degree 6. As a special case of the theorems just stated there results the well known theorem, first proved by Ruffini, that there is no four-valued rational function on five variables. In fact, if such a function were possible there would be a tranisitive substitution group of order 30 and of degree 5. The G, of this group would have to be the symmetric group of degree 3 since there is no other group of order 6 on less than 5 letters. The fact

On multiply transitive groups

The properties of primitive groups that contain transitive subgroups of lower degree were first investigated by JORDAN.t In the Traite des Substitutions he proved that if a pritnitive group of degree nt contains a circular substitution of prime degree (pv) it is at least n p + 1 times The capital importance of this theorem led him to examine the general and much more difficult case, that in which the subgroup of lower degree is merely transiti ve. He obtained the remarkable theorem: t ' If a primitive group G of degree n contains a group F, the substitutions of which displace only 1letters and permute them transitively (p being any integer), it is at least n -p 2q + 3 times transitive, q being the greatest divisor of p such that we can arrange the letters of F in two different ways in systems of q letters which have the property that each substitution of r replaces the letters of each system by those of a single system. If none of the divisors of p have this property (which will happen notably if F is primitive, or formed of the powers of the same circular substitution) G is n p + 1 times transitive. NETTO ? and RUDIO 11 later gave proofs for that special case in which F is primitive. The only other contribution to this theory was made by MARGGRAFF.? He proved that if q is the greatest divisor of p such that the letters of r may be arranged in systems of imprimitivity in at least three different ways, and if p is divisible by some number r such that F admits r + 1 systems of imprimitivity with one letter in common and no two of which have more than one letter in common, G is at least n p 2q + 3 times If these two conditions are not fulfilled, G is n p + 1 times In MARGGRAFF'S

The Sylow subgroups of the symmetric group

In the Sylow theorems t we learn that if the order of a group 9 is divisible by pa (p a prime integer) and not by pa+l, then W contains one and only one set of conjugate subgroups of order pa , and any subgroup of W whose order is a power of p is a subgroup of some member of this set of conjugate subgroups of W. These conjugate subgroups may be called the Sylow subgroups of W. It will be our purpose to investigate the Sylow subgroups of the symmetric group of substitutions. By means of a preliminary lemma the discussion will be reduced to the case where the degree of the symmetric group is a power (pa) of the prime (p) under consideration. A set of generators of the group having been obtained, they are found to set forth, in the notation suggested by their origin, the complete imprimitivity of the group. The various groups of substitutions upon the systems of imprimitivity, induced by the substitutions of the original group, are seen to be themselves Sylow subgroups of symmetric groups of degrees the various powers of p less than pa. They are also the quotient groups under the initial group of an important series of invariant subgroups. In terms of the given notation convenient exhibitions are obtained of the commutator series of subgroups and also of all subgroups which may be considered as the Sylow subgroups of symmetric groups of degree a power of p. Enumerations are made of the substitutions of periods p and pa and the conjugacy relations of the latter set of substitutions are discussed. The subgroup consisting of all substitutions invariant under the main group is cyclic and of order p. The full set of conjugate Sylow subgroups of the symmetric group on pa letters fall into

Scientific Serial

American Journal of Mathernagics, vol. xxiii. No. 3.—Geometry on the cubic scroll of the second kind, by F. C. Ferry, is the conclusion (34 pp.) of a paper commenced in the last number.—Congruent reductions of bilinear forms, by T. J. I'A. Bromwich, contains an account and a slight extension of a method due to Kronecker (Gesamm. Werke, Bd. i. p. 349). This method was employed in the first place for the reduction of two quadratic forms. In the present paper it is applied to four cases of reductions, viz. (1) two symmetric forms (the same as Kronecker's case); (2) a symmetric and an alternate form (3) two alternate forms; and (4) two Hermite's forms. In cases (1)–(3) the substitutions are congruent, while in (4) they are conjugate imaginaries. Mr. Bromwich gives a list of the principal papers which deal with the problems he has considered in his article. On the imprimitive substitution groups of degree fifteen and the primitive substitution groups of degree eighteen, by E. Norton Martin, was presented, in abstract and in a slightly different form, at the summer meeting of the American Mathematical Society in 1899. Herein he has added two new groups to his original list, viz. the groups with five systems of imprimitivity simply isomorphic to the alternating and symmetric groups of degree five, and he mentions that Dr. Kuhn reported at the February (1900) meeting of the Society that he had carried the investigation further by adding twenty-eight to the seventy groups found by Mr. Martin. The list even now does not claim to be absolutely complete, since omissions are always possible. A somewhat long list of recent papers on the subject is appended to the article.—Removal of any two terms from a binary quantic by linear transformations, by Bessie G. Morrison, discusses these linear transformations and gives applications to the non-singular cubic, quartic, quintic and sextic.