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Edu Contemporary Mathematics Volume 524, 2010 Threads Through Group Theory Persi Diaconis Abstract. This paper records the path of a letter that Marty Isaacs wrote to a stranger. The tools in the letter are used to illustrate a different way of studying random walk on the Heisenberg group. This paper also explains how the letter contributed to the development of super-character theory. 1. Introduction Marty Isaacs believes in answering questions. They can come from students, colleagues, or perfect strangers.

This conjecture is known to be true for symmetric groups, general linear groups, and various families of simple groups. Isaacs and Navarro stated and proved a refined version of the Alperin weight conjecture for π-separable groups in [19], and we now briefly discuss their result. (For more discussion of the history of the proof of the Alperin weight conjecture for solvable groups, see [19]). Isaacs and Navarro refine and prove the Alperin weight conjecture using the vertex subgroup. We say a π-weight of a π-separable group is a pair (Q, ϕ), where Q is a π -subgroup of G and ϕ ∈ Irr(NG (Q)/Q) is such that ϕ(1)π = |NG (Q)/Q|π .

S ) is π-factorable. It can then be shown that the stabilizer of γ is proper in G. ) Letting ψ be the Clifford correspondent of χ in the stabilizer subgroup, we then recursively define the normal (resp. subnormal) nucleus of χ to be the normal (resp. subnormal) nucleus of ψ. Notice two important properties of the nucleus pair (U, ρ) constructed above. First, the nucleus character ρ induces irreducibly to χ. Secondly, notice that the construction process terminates when the Clifford correspondent constructed in some step is π-factorable.

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