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By American Mathematical Society

ISBN-10: 0387068406

ISBN-13: 9780387068404

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Extra resources for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

Example text

Proof. Keep the above notation. 1 shows that i apei (Pi ) ≤ pre i i for each i. The first claim now follows from the preceding lemma, and the second is an immediate consequence. 3 Let G be a group such that every finite quotient of G is soluble of rank at most r. Then sn (G) ≤ n2+r for all n. 8. 4 Let G be a finite group. If n = pe11 . . pekk where p1 , . . , pk are distinct primes, then k an (G) ≤ nk · apei (Pi ) i i=1 where for each i, Pi is a Sylow pi -subgroup of G. Proof. Let H be a subgroup of index n in G.

The result now follows because each subgroup H of G is of the form (H ∩ P1 ) × · · · × (H ∩ Pk ). 5 Abelian groups I For certain abelian groups it is possible to estimate the subgroup growth with some precision. 1 If p is a prime then 1− 1 pd pk(d−1) ≤ apk (Z(d) ) ≤ 1+ 1 p−1 for all k ≥ 0. Proof. Suppose apk (Z(d) ) = f (d, k)pk(d−1) d pk(d−1) 22 CHAPTER 1. BASIC TECHNIQUES OF SUBGROUP COUNTING for all d and k. Then f (1, k) = 1 for all k. Now let d > 1. 15) i=0 k f (d − 1, i)pi(d−2) · p(k−i)(d−1) = i=0 k f (d − 1, i) · p−i .

Xk hk ) for any h1 , . . , hk ∈ H, so the number of k-tuples corresponding to distinct subgroups of index n in G is k k at most |G| / |H| = nk . The result follows since H is generated by its Sylow xi subgroups H ∩ Pi (i = 1, . . , k). 5 Let |G| = pb11 . . pbt t where p1 , . . , pt are distinct primes, and put b = max{bi | 1 ≤ i ≤ t}, r = max{rpi (G) | 1 ≤ i ≤ t}. Then an (G) ≤ nν(n)+r ≤ nt+r ≤ nt+b ≤ n2 log|G| for each n, where ν(n) denotes the number of distinct prime divisors of n. 1 to each Pi , and the remaining inequalities are clear (we may assume that n | |G| since an (G) = 0 otherwise).

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A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco by American Mathematical Society


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