All Questions
Tagged with finite-groups lattice-theory
28 questions
3
votes
1
answer
774
views
What's concrete model for Coxeter complexes?
We know for every Coxeter system $(W,S)$ there is a Coxeter complex associated by its cosets of parabolic subgroups. In Wachs's note Poset Topology p.12-13 she mentioned for the Coxeter complex of ...
- 311
6
votes
3
answers
213
views
Groups whose poset of direct factors are lattices
Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...
- 933
6
votes
2
answers
493
views
Finite lattice representation problem checking
[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem ...
1
vote
1
answer
139
views
Is an Eulerian subgroup lattice boolean?
Let $G$ be a finite group and $\mu$ the Möbius function of the subgroup lattice $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ Kg \ | \ K<G, \ g \...
16
votes
1
answer
804
views
Existence of a faithful irreducible representation using Möbius function
Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.
Consider the Euler totient of $G$ defined as follows:
$$ \varphi(G) = \sum_{H \le G}\mu(H,G) |H| $$
Let $X=\{M_1, \...
3
votes
1
answer
173
views
Is there an atom K of [H,G]≃B2 with |K:H|≡|G:H|(mod 2)?
Let $[H,G]$ be a rank $2$ boolean interval of finite groups.
Statement 1: There is an atom $K$ of $[H,G]$ such that $|K:H|≡|G:H|($mod $ 2)$.
The following picture illustrates the statement.
...
2
votes
0
answers
97
views
Is the bounded coset poset of a boolean interval of finite groups, Cohen-Macaulay?
Let $[H,G]$ be a boolean interval of finite groups and let $\hat{C}(H,G)$ be its bounded coset poset (i.e. the poset of cosets $Kg$ with $K \in [H,G]$, bounded below by $\emptyset$ and bounded above ...
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 $...
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, ...
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 $|[...
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 ...
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 \...
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 ...
1
vote
1
answer
116
views
Can the reversed lattice of a subgroups interval be represented?
Let $G$ be a finite group and $H$ a subgroup. The interval $[H,G]$ is the lattice of overgroups of $H$.
It is an open problem to know if every finite lattice can be represented by such an interval (...
0
votes
1
answer
187
views
Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?
Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and $...
- 17
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 \...
4
votes
1
answer
275
views
The groups with symmetric subgroups lattice
Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.
If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the ...
- 1,169
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 ...
10
votes
2
answers
724
views
Can any finite lattice be realized as an intermediate subgroups lattice?
Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...
13
votes
1
answer
2k
views
A dual version of a theorem of Øystein Ore in group theory
This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved:
Theorem: Let $[H, G]$ be a distributive interval of finite groups. Then $\exists g \in G$ such ...
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 $\...
7
votes
1
answer
256
views
Self-duality of the subgroup lattice of $G\times H$
Let $G$ and $H$ be finite groups and $\gcd(|G|,|H|)=1$.
Suppose the lattice of all subgroups of $G\times H$ is self-dual. Is the lattice of all subgroups of $G$ self-dual?
- 1,145
3
votes
1
answer
586
views
a characterization for cyclic groups [duplicate]
Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z_n$. Is $G$ cyclic?
- 1,145
2
votes
1
answer
375
views
generality of the lattice of normal subgroups
Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?
- 1,145
1
vote
0
answers
224
views
Does $[H_i , G_i]$ distributive imply $[H_1 \times H_2, G_1 \times G_2]$ modular?
Let $L(G)$ be the subgroup lattice of $G$ and $[H, G]$ an interval in $L(G)$.
A lattice $(L, \wedge, \vee)$ is distributive if $a∨(b∧c) = (a∨b) ∧ (a∨c)$, $\forall a,b,c \in L $, and is modular if ...
0
votes
2
answers
202
views
Products of maximal inclusions of finite groups with a non-obvious intermediate
Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups.
Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
2
votes
0
answers
199
views
Existence of inclusions of finite groups with a particular lattice property
Definition : Let $\sim$ be the equivalence relation on inclusions of finite groups, generated by :
$(H \subset G) \sim (\phi(H) \subset \phi(G))$, with $ \phi: G \to L$ a finite group morphism and ...
22
votes
0
answers
1k
views
Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?
If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
- 1,241