A 3-local characterization of the group of the Thompson sporadic simple group

Fowler, Rachel Ann Abbott (2007). A 3-local characterization of the group of the Thompson sporadic simple group. University of Birmingham. Ph.D.

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Abstract

In this thesis we characterize the Thompson sporadic simple group by its 3-local structure. We study a faithful completion, $G$ , of an amalgam of type F $_3$ with the property that $N_G(Z(L_\beta)) = G_\beta$ . We first assume no additional 3-local structure and use a $\kappa$ -proper hypothesis to establish that the completion $G$ with this property contains a subgroup $Y$ of order 3 such that $N_G(Y)\cong (3 \times$ G $_2$ (3)) : 2. Secondly, we assume that $G$ contains such a subgroup $Y$ with $N_G(Y)\cong (3 \times$ G $_2$ (3)) : 2 and show that for an involution $t \in G, C_G(t)$ has shape 2 $^{1+8}_+$ .Alt(9). We then invoke a theorem of Parrott to show that $G \cong$ Th.