On the Construction of Normal Subgroups

This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Finally, the character theory of groups with a Sylow p-subgroup of order p is studied. Each chapter concludes with a set of problems. The book is aimed at graduate students, with some previous knowledge of ordinary character theory, and researchers studying the representation theory of finite groups.

MR. D. E. LITTLEWOOD, whose appointment to the chair of mathematics at the University College of North Wales, Bangor, in succession to Prof. T. G. Cowling, has recently been announced, is Well known as an algebraist. His best-known work is concerned with the theory of group characters, a branch of algebra which has been extensively developed in the last fifty years and which has an unusually wide application to fields as diverse as the algebraic theory of invariants and modern quantum mechanics. Littlewood‘s work is in the direct line of tradition associated with the names of Frobenius, Alfred Young, and Schur ; and his book, "The Theory of Group Characters" published in 1940, is a standard work on the subject. In recent papers, Littlewood has developed a calculus of a class of symmetric functions known as Schur functions which have recently been shown to have extensive applications in invariant theory.

REPRESENTATION theory is a subject of many aspects, having important contacts with several branches of mathematics and mathematical physics. Even if text-books were much commoner than they are, there would still be room for an introduction to the subject so admirable as this. The very diversity of the theory demands a specialized outlook from those who write concerning it, but within the limits that implies, the account given by Prof. Littlewood is excellent. The Theory of Group Characters and Matric Representations of Groups By Dudley E. Littlewood. Pp. viii + 292. (Oxford: Clarendon Press; London: Oxford University Press, 1940.) 20s. net.

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

Irreducible characters and normal subgroups in groups of odd order

Persi Diaconis and I. M. Isaacs generalized the character theory to super-character theories for an arbitrary finite group (Diaconis and Isaacs, in Trans Am Math Soc 360(5):2359–2392, 2008). In these theories, the irreducible characters are replaced by certain so-called supercharacters, and the conjugacy classes of the group are replaced by superclasses. Also, Diaconis and Isaacs discussed supercharacter theories and gave some properties of them. We consider in this note certain sums of irreducible Brauer characters and compatible unions of regular conjugacy classes in an arbitrary finite group and we give a generalization of the Brauer character theory to super-Brauer character theories. We also discuss super-Brauer character theories and obtain some results which are similar to those of Diaconis and Isaacs.

This paper is about the situation where χ is an irreducible character of a finite group G and K is a normal subgroup. A construction of Serre's relating the characters of G with those of G/K is used to give a new proof of a well-known lemma concerning the case that χ \κ is irreducible and to generalize this lemma. It is seen that the irreducibility of χ I* is equivalent to the property that (1/| K\) ΣX^K I χ(x) I2 = 1 for each coset of G modulo K and also to the property that χ is not a component of λχ for any irreducible character λ of G/K except for λ — 1. The subgroup Jx = Ji(χ) is defined as the intersection of the kernels of the irreducible characters λ of G/K for which χ is a component of λχ. It is seen that an irreducible component σ of the restriction of % to if will extend to Ju eJχ(γ) = eκ(χ) and Jx is the maximal normal subgroup with these two properties.

Let G be a finite metabelian group with all nonlinear irreducible characters rational.Then the exponent of the commutator group G' is a prime or divides 16, 24, or 40.If G' is also cyclic, then its order is a prime or divides 12.1. Introduction.Let G be a finite group having all characters (over the complex field) rational.Markel [6] proved that if G is supersolvable then G has order \G\ = 2*33.In [4] Gow proved that if G is solvable then \G\ = 2'3S5\ and in [7] Vishnevetskij proved similar results if G is finite with some added conditions.In this paper we assume that G is a finite metabelian group having all nonlinear (absolutely) irreducible characters rational.After proving a few preliminary results we prove, in 3, that the exponent of G' is either a prime or divides 16, 24, or 40.Assuming that G' is cyclic we prove, in 4, that \G'\ is a prime or it divides 12.In particular if all the characters of G are rational, then the results of this paper imply that \G\ = 2<3a.2. Preliminary results.Let G be a finite group.By a character of G we shall always mean a character over the complex field, and by an irreducible character we shall always mean an absolutely irreducible character.A character x of G is rational (real) if x(q) is rational (real) for all g E G.If K is a normal subgroup of G and a E G, then (a) and (aK) denote the cyclic subgroups of G and G/K respectively, and |a|n and \aK\o denote the orders of (a) and (aK) respectively.Also exp(G) denotes the exponent of G.If A is a group that acts on a group B then CA(B) = {x E A\bx = b for all b E B}.LEMMA 1.Let G be a finite group, K a normal subgroup ofG, x a linear character ofK such that xG is irreducible, L = kerx, and N(L) the normalizer ofL in G. Then xG is real if and only if some element of N(L) inverts every element of K/L and xG S rational if and only if N(L)/K ~ Aut(K/L), the automorphism group of K/L.PROOF.Let xi be the Q-conjugate (complex conjugate) of x-Then kerxi = L, Xi is irreducible and is Q-conjugate (complex conjugate) to xG-From [3, (45.5)],XG = XG if and only if Xi and x are G-conjugate.The results follow.A somewhat deeper result concerning the realness of xG is given in [2, 2].Now assume G is a finite metabelian group, A a normal subgroup of G such that A and G/A are abelian.Let x be a linear character of A with ker x = L, and K(L) a subgroup of N(L) containing A such that K(L)/L is a maximal abelian subgroup

Extension groups between simple Mackey functors

Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of G is equal to the order of G. N. Snyder proved that if |G| = d(d + e), then the order of G is bounded in terms of e, provided e &gt; 1. Y. Berkovich proved that in the case e = 1 the group G is Frobenius with the complement of order d. We study a finite nontrivial group G with an irreducible complex character Θ such that |G| ≤ 2Θ(1)2 and Θ(1) = pq, where p and q are different primes. In this case we prove that G is solvable groups with abelian normal subgroup K of index pq. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime p and of order less than 2p4 is isomorphic to L2(q), L3(q), U3(q), Sz(8), A7, M11 or J1.

Cohomology of Semidirect Product Groups

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

Let G be a group, N be a normal subgroup of G, and p be an irreducible (finite-dimensional) character of N in some algebraically closed field a. Assume that the stabilizer G, of p in G has finite index in G. Then Clifford's theory [2] gives us a central extension G of the multiplicative group F of a by GIN together with a one-to-one correspondence between the set Ch (G I A) of all irreducible characters of G having p as an N-constituent and the set Ch (G ) of all projective irreducible characters of GjIN corresponding to the extension G , i.e., of all irreducible characters of G having the natural embedding of F in ! as F-constituents. Now we add another normal subgroup K of G containing N and an irreducible character T of K having p as an N-constituent. We may take K to be the inverse image in G of Ky/N. Then K is a normal subgroup of G . Let T e Ch (K I q) correspond to * e Ch (K ) under the Clifford correspondence. One easily verifies that Ch (G I A) c Ch (G I9) corresponds to Ch(G I A). Furthermore, the stabilizers G and G, are related by (0.la) G o is the inverse image in G of (G , G,)/N. (0. lb) G, = (G, n G,)K. It follows that [G : G +] is finite if and only if [G: G,] is. When this happens, we may apply Clifford's theory to G, K, and A , obtaining an extension G of F by G,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). We may also apply Clifford's theory to G , K , and *, obtaining an extension G of F by G ,/K and a one-to-one correspondence between Ch (G I A) and Ch (G ). By (0.1) there is a natural isomorphism of G ,/K (Gfl GW)/K? onto G,/K. So G and G are extensions of F by isomorphic groups. Furthermore both

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then $${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality $${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if $${{\rm cd} {(G \mid N')}}$$ is a set of non-trivial p–powers for some fixed prime p.

Self-dual modules in characteristic two and normal subgroups