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By Mark L. Lewis, Gabriel Navarro, Donald S. Passman, Thomas R. Wolf

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The final section outlines an approach to the open problem of extending the analysis to the group of n × n uni-upper-triangular matrices with coefficients in Cp . This leads to the study of super-characters, a subject developed in later work with Marty. What is the effect of kindness to strangers? This paper follows one of those threads. I think our subject is woven from these. Marty has spun out hundreds of threads which lead to all corners of group theory. We are in his debt. 2. Random walk on finite groups Introductions to random walk on finite groups appear in [5, Chapt.

8] Suppose that G is a group of odd order with normal subgroup N , and let Q be a p-subgroup of G and P = Q ∩ N . If ϕ ∈ IBrp (G|Q) and θ ∈ IBrp (N |P ), then ϕ lies over θ if and only if ϕ1 ∈ IBrp (NG (P )|Q) lies over θ ∈ IBrp (NN (P )|P ). Thus for a fixed character θ ∈ IBrp (N |P ), the map ϕ → ϕ1 is a bijection from {ϕ ∈ IBrp (G|Q) | [ϕN , θ] = 0} to {ψ ∈ IBrp (NG (P )|Q) | [ψNN (P ) , θ] = 0}. It turns out that actually more is true. In [5], it is shown that with the above notation, [ϕN , θ] = [ψNN (P ) , θ].

Uniqueness of vertex pairs of lifts in groups of odd order. Recall that if χ ∈ Irr(G) is such that χo = ϕ ∈ Iπ (G), then we say that χ is a lift of ϕ. Much of the study of lifts of Brauer characters can be characterized by seeking answers to the following question: What properties of Brauer characters are inherited by their lifts? In this section we will see the first of two results that show that if G has odd order, then the lifts of ϕ ∈ Iπ (G) behave like ϕ. Our first important yet relatively easy result shows that if G has odd order, and χ ∈ Irr(G) is a lift of ϕ ∈ Iπ (G), then the vertex pairs of χ all have linear vertex characters.

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