Summary of Some Research Findings From Early Fall 2007 Nicholas Zoller Lehigh University Let G n,d be the n th group in the Magma/GAP database of transitive groups of degree d . Consider the minimal transitive group G 24 , 1489 found in the list in . The center of G 24 , 1489 is trivial, so G 24 , 1489 cannot be the Galois group of a CM-ﬁeld. However, by looking at the minimal partitions of G 24 , 1489 , we can determine if it serves as a witness that another group of order 2 · | G 24 , 1489 | = 2 · 576 = 1152 is ρ-minimal. (See  for a deﬁnition and discussion of ρ-minimality). Magma veriﬁes that G 24 , 1489 has three minimal partitions, each consisting of 12 sets of imprimitivity of size 2. Call them P 1 , P 2 , and P 3 . Magma can ﬁnd the image and kernel of the action G 24 , 1489 on each of these block systems. Let K i and I i be the kernel and image, respectively of each of these actions, for i = 1 , 2 , 3 . Magma calculates that
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This note was uploaded on 02/29/2008 for the course MATH 205 taught by Professor Zhang during the Spring '08 term at Lehigh University .