Relative normal complements and extendibility of characters

Introduction. All groups in this paper are assumed to be finite. Let G be a group with subgroups H 0 and H where H o A H, then a subgroup Go of G is called a relative normal complement in G of H over H o if Go & G, G = Go H and H o = G o c~ H. Given a group G with subgroups H o and H where H o/~ H, many theorems have been proved which guarantee the existence of a relative normal complement G O of H over H o. In the work of Brauer [1], Suzuki [10], Dade [2], Leonard [8], Leonard and McKelvey [7], Sah [9] and others, the proof of the existence of a relative normal complement depends on showing that certain generalized characters of H can be extended to generalized characters of G. In this paper, we present theorems which explicitly relate the extendibilitiy of certain irreducible characters of H to G and the existence of relative normal complements. Indeed, if Go is a relative normal complement of H over H o, then G/G o H/Ho. Thus, every character of H having H o in its kernel can be extended to G. Several of the theorems in this paper may be viewed as partial converses to this observation. Before, these theorems can be stated, some notation and terminology is necessary. Let n be a set of primes and denote the complementary set of primes by ~'. For a finite group G and prime p, let 1@1 denote the order of a Sylow p subgroup of G. Then [G 1~ is defined by [ GI~ = I~ I @1. We say that G is a n-group if [ G [ = I G[.. A subgroup K of G is a Hall pET~ re-subgroup of G if IKI = IKI~ --IGI=. An element x in G is a n-element if (x) is a n-group. Every element x in G has a unique decomposition x = x,~x~, = x,~, x~ into a n-element x. and a n'-element x~,. Further, x~ and x., are powers of x. If x and y are elements of a subgroup K of G, then x and y belong to the same n-section of K if their n-parts x~ and y. are conjugate in K. If S is a subset of G, then S a'~ denotes the union of all n-sections of G which intersect S. For any non-empty set A we let ]A[ denote the number of elements in A. IfH and H o are subgroups of G with Ho A H and rc is the set of primes dividing (H : Ho), then the existence and uniqueness of a relative normal complement have been related by Leonard [8] and others [4] to the equality I(H - Ho) G' "1 = (G : H) IH - Hol. This equality plays a similar role in several theorems below. If K is a group, let Irr K denote the set of irreducible characters of K. If K o is a normal subgroup of K, let Irr (K/Ko) denote the set of irreducible characters of K whose kernels contain Ko. If)l and B are disjoint sets, then A ~ B denotes the disjoint union of A and B. We may now state the theorems to be proved.