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By Pannenberg M.

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2. The result follows, with b = c + 1. 8 to soluble groups in several of the above results. It is also useful for dealing with groups having non-abelian composition factors. Denote by µ(G) the minimal index of a proper subgroup of G; of course µ(G) = |G| if G is cyclic of prime order, but there are also good lower bounds for µ(G) when G is a non-abelian simple group. 9. 1 Suppose that 1 = Gk Gi−1 /Gi and suppose that Gk−1 29 ... G0 = G. Put Qi = µ(Qi )c ≥ |Qi | rk(Qi ) ≤ r for each i, where (c − 1)r ≥ 2.

Pm − pm−1 )pm(n−m) > κ(p)−1 pnm . n Each of these corresponds to |Φ(H)| = pn(n−m) generating n-tuples for H, so 2 H has more than κ(p)−1 pn ordered generating sets of size n. Now the number of n-tuples of elements in G is pdn ; consequently the number of subgroups of order pn in G is less than pdn = κ(p)p(d−n)n . κ(p)−1 pn2 This gives the result on putting n = d − r. 17) p in particular ap (G) ≥ pd−1 . 6. 19) where log(p − 1) ; log p note that 0 < µ(p) < 1 and that µ(p) → 1 as p → ∞. 2 Let G be a finite p group.

28) is equal to 2 min(ei , fj ) + 2 min(er , fs ) + 2 i

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