All Questions
Tagged with lattices finite-groups
20 questions
37
votes
2
answers
2k
views
A group-theoretic perspective on Frankl's union closed problem
Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
- 85.6k
16
votes
2
answers
992
views
Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
- 2,179
12
votes
2
answers
832
views
24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise
Two vectors from Leech lattice - as defined on wikipedia - have scalar product $\pm 32,\pm 16, \pm 8$ or $0$. Do there exist 24 vectors having scalar product 8 pairwise ? When we consider unit vectors ...
user21230
10
votes
1
answer
648
views
Octonionic reflection groups
Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008).
There are 819 E8 sublattices defined by
$ (2\lambda, 0, 0); $
$ (\lambda \overline{s}, (\lambda \...
user21230
9
votes
6
answers
683
views
Which finite nonabelian groups have long chains of subgroups as intervals in their subgroup lattice?
Given N, what is a finite non-abelian (and preferably non-solvable) group G in whose subgroup lattice Sub[G] there is an interval that is a chain of length at least N?
Since N can be arbitrarily ...
- 1,241
7
votes
2
answers
700
views
What groups have a second maximal subgroup below exactly four maximal subgroups?
I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...
- 1,241
7
votes
1
answer
565
views
Are the distributive permutation groups linearly primitive?
An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no non-...
6
votes
1
answer
395
views
Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?
I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24:
Now let us recall that the ...
- 22.3k
6
votes
1
answer
278
views
What is the automorphism group of the tensor square of the Leech lattice?
The tensor square of the Leech lattice is an even unimodular lattice of dimension 576 which, unless I am very mistaken, has no roots. Its automorphism group contains a group of shape $2 \cdot \mathrm{...
- 54.6k
5
votes
1
answer
165
views
Can any finite distributive weighted lattice be realized by inclusion of groups?
By theorem 2.1 here, any finite distributive lattice $\mathcal{L}$ can be realized as an intermediate subgroups lattice.
A weighted lattice $(\mathcal{L},\tau)$ is a lattice $\mathcal{L}$ with a ...
5
votes
0
answers
305
views
Are the homogeneous single chain subfactors, Dedekind?
Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
3
votes
0
answers
302
views
What's the ratio of inclusions of finite groups with a distributive lattice?
Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...
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
2
votes
2
answers
199
views
Permutation covering of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...
- 14.1k
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 ...
2
votes
0
answers
123
views
Finite subgroups (lattices) in the large N limit of SU(N)
I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more ...
- 21
1
vote
0
answers
262
views
A question on the poset of classes of isomorphic subgroups of finite groups
Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where
$[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining
$$[...
1
vote
1
answer
259
views
Existence of homogeneous single chain compositions of a given maximal subfactor?
All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a single ...
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 ...
0
votes
1
answer
218
views
Map homemade Leech lattice to classic one
In following question on MathOverflow I received construction of new Leech lattice provided by Noam Elkies. Let's call it $(E)$. This Leech lattice has nice feature that there is easy to see $24$ ...
user21230