![]() $$ We also show that these quotients classify supersingular elliptic curves in characteristic $p$. The normal subgroup Cout(~)(Syl,(G)) of Out(G) is nilpotent, forįor positive integers $1\leq i\leq k$, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon $q$-series $$ \prod_(3 n-2). Evidently Cout(G)(Syl, (G)) is a normal subgroup of Out (G), and the above theorem yields the Corollary. As a special case, we may let M* be 1 and H* be the subgroup Cout(m(Syl, (G)) of all outer automorphisms of G "centralizing" every Sylow subgroup of G. In the situation of the above theorem, the class of the nilpotent group H*/M* is smaller than a function (probably linear) of the number of distinct primes dividing IG[. However, it is likely that it could be improved along the lines.of the Conjecture. It is obvious that this is, in a certain sense, a best possible result. Conversely, given a finite nilpotent group N, there is a finite abelian group G and a section H*/M* of Out(G) "centralizing" every Sylow subgroup of G such that the group H*/M* is isomorphic to N. If G is a finite group, then any section H* /M* of Out (G) "centralizing" every Sylow subgroup of G is nilpotent. We shall prove (in Corollary (3.8) and Theorem (4.15) below) the Theorem. We are interested in sections H*/M* of Out(G) which "centralize" every p-Sylow subgroup of G for every prime p dividing the order [GI of G. We shall say that such a section H*/M* "centralizes" a subgroup S of G if the usual centralizer CAut(G)(S) of S in Aut(G) covers H/M, i.e., if each element a of H*/M* is the natural image of some automorphism c~eH centralizing S. So any section H*/M* of Out(G) (i.e., any factor group of a subgroup H* of Out (G) by a normal subgroup M* of H*) is naturally isomorphic to the section HIM of Aut(G) formed by the inverse images H, M of H*, M*, respectively. The outer automorphism group Out(G) of a finite group G is just the factor group Aut(G)/In(G) of the full automorphism group Aut(G) of G by the normal subgroup In (G) of all inner automorphisms of G.
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