A Group-theoretic Characterization of M-groups
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I. Martin Isaacs, among others, has posed the problem of finding a purely group-theoretic characterization of M-groups which traditionally are defined via character theory. This thesis seeks to understand and answer Isaacs’ question by finding a characterization of M-groups in purely group-theoretic terms. Such a characterization based on cyclic sections of the group G and the irreducible, monomial characters that proceed naturally from them does exist, and was first described by Alan Parks. He describes M-groups based on the notion of good pairs and an equivalence relation on them. If me? is the number of classes of good pairs and nG is the number of rational conjugacy classes for a group G, then mG = nG if and only if G is an M-group.
In order to fully understand this group-theoretic characterization of M-groups, the traditional way of defining M-groups by induced characters had to be explored. First, fundamentals of representation and character theory (including irreducible and induced representations and characters, inner products of characters, the Mackey Theorems and the calculation of character tables) were researched. Next, M-groups themselves were studied, including many specific examples of well-known groups. At this point it was possible to understand the traditional definition of an M-group as a group in which every irreducible character is induced from a linear character of some subgroup of the group. Before attempting to study Parks’ group theoretic characterization, some background preparation in field theory and Galois theory was done. Finally Parks’ article was studied and his characterization was validated.
The conclusion is that there is a verifiable group-theoretic characterization of M-groups. This alternative definition of M-groups serves to add to the body of knowledge about M-groups and how they behave.
CitationPierson, J. (2004). A group-theoretic characterization of M-groups (Unpublished thesis). Texas State University-San Marcos, San Marcos, Texas.
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