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  In mathematics, a prop group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a pgroup. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a prop group to be the inverse limit of an inverse system of discrete finite pgroups. The bestunderstood (and historically most important) class of prop groups is the padic analytic groups: groups with the structure of an analytic manifold over elements.

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  In mathematics, a prop group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a pgroup. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a prop group to be the inverse limit of an inverse system of discrete finite pgroups. The bestunderstood (and historically most important) class of prop groups is the padic analytic groups: groups with the structure of an analytic manifold over such that group multiplication and inversion are both analytic functions.The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the padic numbers, shows that a prop group is padic analytic if and only if it has finite rank, i.e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements. The Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. LeedhamGreen. Theorem D is one of these theorems and asserts that, for any prime number p and any positive integer r, there exist only finitely many prop groups of coclass r. This finiteness result is fundamental for the classification of finite pgroups by means of directed coclass graphs.

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