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Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup} (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size \emph{which is either a prime power or square-free}., 16 pages - revised according to referee's report

Suppose that G is a finite p -solvable group such that N G ( P ) /P has odd order, where P ∈ Syl p ( G ). If χ is an irreducible complex character with degree not divisible by p and field of values contained in a cyclotomic field Q p a , then every subnormal constituent of χ is monomial. Also, the number of such irreducible characters is the number of N G ( P )-orbits on P/P’ . Introduction There are few results guaranteeing that a single irreducible complex character χ ∈ Irr( G ) of a finite group G is monomial. Recall that χ ∈ Irr( G ) is monomial if there is ƛ ∈ Irr( U ) linear such that ƛG = χ . It is known that every irreducible character of a supersolvable group is monomial, for instance, but this result depends more on the structure of the group rather than on the properties of the characters themselves. An exception is a theorem by R. Gow of 1975 ([3]): an odd degree real valued irreducible character of a solvable group is monomial. Recently, we gave in [8] an extension of this theorem which also dealt with the degree and the field of values of the character. (Yet another similar monomiality criterium was given in [9]: if the field of values Q( χ ) of χ is contained in the cyclotomic field Q n and ( χ (1) , 2 n ) = 1, then χ is monomial whenever G is solvable.) In this note, we apply non-trivial Isaacs π -theory of solvable groups to give a shorter proof of the above result at the same time that we gain some new information about the subnormal constituents of the characters, among other things. It does not seem easy at all to prove these new facts without using this deep theory. Recall that for every solvable group and any set of primes π , M. Isaacs defined a canonical subset B π ( G ) of Irr( G ) with remarkable properties ([4]). Since, by definition, every χ ∈ B π ( G ) is induced from a character of π -degree, it is clear that B π -characters of π’ -degree are monomial.

In this survey paper we discuss some topics from the theory of subgroup lattices. After giving a general overview, we investigate the local structure of subgroup lattices. A major open problem asks if every finite lattice occurs as an interval in the subgroup lattice of a finite group. Next we investigate laws that are valid in normal subgroup lattices. Then we sketch the proof that every finite distributive lattice is the normal subgroup lattice of a suitable finite solvable group. Finally, we discuss how far the subgroup lattice of a direct power of a finite group can determine the group. Introduction This survey paper is the written version of my four talks given at the Groups – St Andrews 2001 in Oxford conference. I selected some topics on subgroup lattices and normal subgroup lattices according to my personal taste and interest. These topics, of course, cannot cover all interesting and important parts of the theory. For a more complete overview the reader should consult the small book of Michio Suzuki [60] from 1956 and the more recent monograph by Roland Schmidt [54]. The latter one is a thick volume of 541 pages including 384 references. So it is clearly impossible to give a comprehensive survey here. My choice of topics was partly guided by the review of Schmidt's book by Ralph Freese [13].

The main purpose of this paper is to exhibit a doubly-infinite family of examples which are extensions of a p-group by a p′-group, with the action satisfying some conditions of Zappa (1951), arising from his study of dual-standard (meet-distributive) subgroups. The examples show that Zappa's conditions do not bound the nilpotency class (or even the derived length) of the p-group. The key to this work is found in closely related conditions of Hartley (published here for the first time). The examples use some exceptional relationships between primes.

In this paper we formulate a new conjecture and introduce methods to verify it in many cases.

Let G be a finite group and H be a subgroup of G . Then a representation of the group G automatically describes a representation of the subgroup H of G . Such a representation is called a representation of H subduced by a representation of G . Conversely, from a given representation of a subgroup H of G we can form a representation of the group G . Such a representation is called a representation of G induced by a representation of its subgroup H . The problem is to form the irreducible representations (irreps in short) of G from the irreps of its subgroup H . If the group G is finite and solvable (see Section 8.4.1), the problem of forming the irreps of G may be solved by a step-by-step procedure from the trivial irrep of the trivial identity subgroup. This method is possible, for example, for a crystallographic point group. An alternative approach is via the induced irreps of G from the so-called small representations of the little groups of the irreps of H . As a preparation, we shall discuss subduced representations first. Subduced representations Let G = { g } be a group and H = { h } be a subgroup of G . Let Γ( G ) = {Γ( g ); g ∈ G } be a representation of G , then it provides a representation of H by {Γ( h ); h ∈ H }. This representation is called the subduced representation of Γ(G) onto H or the representation of H subduced by the representation Γ( G ).

Let Sylp(G) denote the set of Sylow p-subgroups of a finite group G. We say that G has Property Ip if, for any S∈Sylp(G) and U, V[les ]G, S∩U∈Sylp(U) and S∩V∈Sylp(V) implies S∩U∩V∈Sylp (U∩V). Below, every group is assumed to be finite.

Part I. Preliminary Results: 1. Notation and elementary properties of solvable groups 2. General results on representations 3. Actions of Frobenius groups and related results 4. p-Groups of small rank 5. Narrow p-groups 6. Additional results Part II. The Uniqueness Theorem: 7. The transitivity theorem 8. The fitting subgroup of a maximal subgroup 9. The uniqueness theorem Part III. Maximal Subgroups: 10. The subgroups Ma and Me 11. Exceptional maximal subgroups 12. The subgroup E 13. Prime action Part IV. The Family of All Maximal Subgroups of G: 14. Maximal subgroups of type p and counting arguments 15. The subgroup Mf 16. The main results Appendix Prerequisites and p-stability.

In Part I we saw that the main idea in the construction of a paradoxical decomposition of a set was to first get such a decomposition in a group acting on the set, and then transfer it to the set. An almost identical theme pervades the construction of invariant measures on a set X acted upon by a group G . If there is a finitely additive, left-invariant measure defined on all subsets of G , then it can be used to produce a finitely additive, G -invariant measure defined on all subsets of X . Such measures on X yield that X (and certain subsets of X ) are not G -paradoxical. It was von Neumann [246] who realized that such a transference of measures was possible, and he began the job of classifying the groups that bear measures of this sort. In this chapter we first study some properties of the class of groups having measures and show that it is fairly extensive, containing all solvable groups. We then give the important application to the case of isometries acting on the line or plane, obtaining the nonexistence of Banach-Tarski-type paradoxes in these two dimensions. Definition 10.1. If, for a group G, µ is a finitely additive measure on P(G) such that µ(G) = 1 and µ is left-invariant (µ(gA) = µ(A) for g ∈ G, A ⊆ G), then µ will be called simply a measure on G .