All Questions
11 questions
2
votes
3
answers
888
views
The number of submodules of $\mathbb{Z}_q^n$
Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...
- 503
9
votes
1
answer
650
views
A stronger version of a problem of Kenneth Brown using representations
Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$.
The reduced Euler characteristic of the order complex of the coset poset $\{ ...
- 1,388
3
votes
1
answer
193
views
Maximal factorization of finite simple groups and no extra intermediate
The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
- 44.4k
8
votes
0
answers
435
views
A relation between intersection and product on Boolean interval of finite groups
Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
6
votes
0
answers
1k
views
Frobenius formula
I know two formulas by the name of Frobenius.
The first one computes the number
$$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$
where $...
- 63.9k
12
votes
0
answers
191
views
Non-Boolean Eulerian interval of finite groups
An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following:
It ...
4
votes
1
answer
171
views
Are descents in alternating subgroup counted by $h$-vector?
Consider the alternating subgroup $A_n$ of the symmetric group $S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $h$-vector $h_i$ equals the number of elements of $A_n$ with ...
- 63.9k
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 \...
13
votes
1
answer
651
views
The Möbius number of the nonabelian finite simple groups
Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(...
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.
...
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