Allsup, John David (2007)
Ph.D. thesis, University of Birmingham.
 AbstractThis thesis concerns Mfinite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary nonMfinite sets via suprema and infima of appropriate Mfinite sets. The basic properties of the measures are covered, along with nonmeasurable sets and the use of endextensions. Next we look at nonstandard finite permutations, introducing nonstandard symmetric and alternating groups. We show that the standard cut being strong is necessary and sufficient for coding of the cycle shape in the standard system to be equivalent to the cycle being contained within the external normal closure of the nonstandard symmetric group. Subsequently the normal subgroup structure of nonstandard symmetric and alternating groups is given as a result analogous to the result of Baer, Schreier and Ulam for infinite symmetric groups. The external structure of nonstandard cyclic groups of prime order is identified as that of infinite dimensional rational vector spaces and the normal subgroup structure of nonstandard projective special linear groups is given for models elementarily extending the standard model. Finally we discuss some applications of our measure to nonstandard finite groups.

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