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By Glenn O. E.

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Yt } to the right cosets of D in N . If H is a subgroup of G with H ∩ N = D then H= Dyf (i) xi i∈S(H) for some subset S(H) of {1, . . , q} and some function f : S(H) → {1, . . , t}. Now |G : H| = tq/ |S(H)| , so if |G : H| ≤ n then either t < n or t = n and S(H) = {1, . . , q}. Suppose that t < n. Then putting f (i) = ∞ for each i ∈ / S(H), we see that H is determined by a function from {1, . . , q} to {1, . . , t, ∞}; so the number of such subgroups H is at most (t + 1)q ≤ nq . On the other hand, if t = n then H is determined by a function from {1, .

Slightly more sophisticated methods will also appear, which involve restricting to Sylow subgroups or to soluble subgroups. We give a fairly thorough account of subgroup-counting in abelian groups; this is essential because the most usual way to obtain lower bounds for sn (G) in a general group G is to locate an elementary abelian section A ‘near the top’ of G and then relate sn (G) to sn/m (A), where m is the index of A in G. We recall some notation: d(G) : the minimal cardinality of a generating set for G (topological generating set if G is profinite); rk(G) = sup {d(H) | H a finitely generated subgroup of G} ; rp (G) = sup {d(H) | H a p-subgroup of G} (when G is finite); an (G) : the number of subgroups of index n in G (open subgroups if G is profinite); ∞ n sn (G) = aj (G), s(G) = aj (G) j=1 j=1 11 12 CHAPTER 1.

9). Suppose finally that G is finite. If H ≤ G and H ∩ N = D then H/D ∼ = HN/N ≤ Q so H is generated by D together with at most rk(Q) further elements. 11) follows. In part (iii) of the last proposition we bounded s(G) in terms of s(N ) and rk(Q). 3 Suppose that G is finite and that N is soluble, of derived length l and rank r. Then 3r 2 +lr s(G) ≤ s(Q) |N | lr 3r 2 +lr |Q| ≤ s(Q) |G| . 4 If Q is finite and A is a finite Q-module of (additive) rank r then 3r 2 −1 r H 1 (Q, A) < |A| |Q| . Proof.

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An Algorism for Differential Invariant Theory by Glenn O. E.


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