The number of subgroups of a finite group (II)

Abstract

Let A be a finite group, and let p be a prime. Suppose that p^s is the highest power of p dividing $|A/A\prime |$, where $A\prime$ is the commutator subgroup of A, and that the type $\lambda =({\lambda }_1, {\lambda }_2,\dots )$ with ${\lambda }_1\ge {\lambda }_2\ge \cdots$ of a Sylow p-subgroup of $A/A\prime$ satisfies either ${\lambda }_2\le 1$ or ${\lambda }_2=2$ and ${\lambda }_3=0$. Let $m_A(d)$ denote the number of subgroups of index d in A. If $1\le i\le [(s+1)/2]$ and q is a positive integer such that $\gcd (p,q)=1$, then $m_A(q{p}^{i-1})-m_A(q{p}^i)$ is a multiple of pⁱ and $m_A(q{p}^{[(s+1)/2]})-m_A(q{p}^{[(s+1)/2]+1})$ is a multiple of $p^{[s/2]}$.

Keywords:

• Complement
• finite p-group
• homomorphism
• semidirect product
• subgroup

2000 Mathematics Subject Classification:

• Primary 20B05
• Secondary 20D15
• 20D40
• 20E07