All Questions
9 questions
13
votes
2
answers
2k
views
Generalization of a theorem of Øystein Ore in group theory
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 $\...
- 2,183
6
votes
1
answer
2k
views
Generalization of the fundamental theorem of cyclic groups
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 ...
- 2,821
4
votes
4
answers
485
views
What are the rank 3 boolean intervals [H,G], with G simple group?
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 $...
- 2,773
9
votes
1
answer
235
views
Is a boolean interval of finite groups linearly primitive?
Let $[H,G]$ be an interval of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ , \forall h \...
5
votes
0
answers
95
views
Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?
Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice.
Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$.
There is an OEIS page for the sequence $s(n)$: A018216
1, 2, 2, 5, 2, ...
CommunityBot
- 1
2
votes
2
answers
365
views
A good upper-bound for the cardinal of an interval of finite groups
This post is a relative version of General bound for the number of subgroups of a finite group
Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.
Question: What is a good upper-bound of $|[...
CommunityBot
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2
votes
0
answers
154
views
Nonvanishing of the dual Euler totient on boolean intervals of finite groups
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 ...
CommunityBot
- 1
6
votes
1
answer
629
views
Positivity of the alternating sum of indices for boolean interval of finite groups
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 ...
CommunityBot
- 1
1
vote
0
answers
81
views
An optimal lower bound related to generators in a boolean interval of finite groups
Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg \...
CommunityBot
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