All Questions

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and $\...

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...

Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows: Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order. proof: see ...

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$. Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...