Let A be a finite group, and let p be a prime. Suppose that ps is the highest power of p dividing |A/A'|, where A' is the commutator subgroup of A, and that the type λ=(λ1, λ2,…) with λ1≥λ2≥⋯ of a Sylow p-subgroup of A/A' satisfies either λ2≤1 or λ2=2 and λ3=0. Let mA(d) denote the number of subgroups of index d in A. If 1≤i≤[(s+1)/2] and q is a positive integer such that gcd(p,q)=1, then mA(qpi−1)−mA(qpi) is a multiple of pi and mA(qp[(s+1)/2])−mA(qp[(s+1)/2]+1) is a multiple of p[s/2].
This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Algebra on 22/02/2019, available online: http://www.tandfonline.com/10.1080/00927872.2018.1527918
The number of subgroups of a finite group (II)
CA_47_5_1964_1972
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