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7
votes
4answers
380 views

Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
11
votes
1answer
633 views

Are there infinitely many natural numbers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ f_1(n_1, m_1) = 30n_1m_1 + 23n_1 + 7m_1 + 5\\ f_2(n_2, m_2) = 30n_2m_2 + 17n_2 + 13m_2 + 7\\ f_3(n_3, m_3) = 30n_3m_3 + 23n_3 + 11m_3 + 8\\ f_4(n_4, m_4) = ...
6
votes
1answer
127 views

Strongly minimal covers

Let $H=(V,E)$ be a hypergraph, that is $V$ is a set and $E\subseteq \mathcal{P}(V)$. We say that $C\subseteq E$ is a cover of $H$ if $\bigcup C = V$. A cover $M\subseteq E$ is said to be strongly ...
4
votes
1answer
179 views

Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.
2
votes
1answer
25 views

Maximal expansions of strongly minimal covers of hypergraphs

Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ ...