Intereting Posts

if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
A bounded holomorphic function
Solvability of a system related to the subsets of {1,2,3}
Problem book on differential forms wanted
Bijection between an infinite set and its union of a countably infinite set
Computing second partial derivative with polar coordinates
The exceptional Klein four group
Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
Showing that Q is not complete with respect to the p-adic absolute value
Calculate sums of inverses of binomial coefficients
Size of Jordan block
The limit of sin(n!)
Rules for Product and Summation Notation
Construction of a Hausdorff space from a topological space
What happens if you repeatedly take the arithmetic mean and geometric mean?

**My Definition:** The finite sequence $x_1,x_2,…,x_m$ of nonnegative integers is said to be *generated by a finite group* $G$ iff

- $n:=|G|=x_1+x_2+\cdots+x_m$.
- $n$ has $m$ divisors.
- if $d_1<d_2<\cdots<d_m$ are all divisors of $n$, for every $k = {1,…,m}$ we have

$$x_k=\left|\{x \in G \mid o(x)=d_k\}\right|.$$

What are all conditions on a finite sequence $x_1,x_2,…,x_m$ such that it is generated by some group $G$?

Some conditions I found:

- Quadratic Integers in $\mathbb Q$
- Characterization of short exact sequences
- Rings with $a^5=a$ are commutative
- $M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?
- An example of a division ring $D$ that is **not** isomorphic to its opposite ring
- Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

- For $k = {1,…,m}$ we have $$\phi(d_k)\mid x_k$$ where $m$ and $d_k$ are as above.
- $x_1=1$.
- $x_m\leq\phi(n)$ and if $x_m=\phi(d_m)$ then for $k = {1,…,m}$, $$x_k=\phi(d_k) $$
- If $x_k\ne 0$ then for every $i$ such that $d_i|d_k$, we have $$x_i\ne 0$$
- If $d_k$ is a prime, $$x_k\ne 0$$

- Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
- Quotient rings of Gaussian integers
- The ideal $I= \langle x,y \rangle\subset k$ is not principal
- What is the quotient ring $\Bbb Z/(2x-3)$?
- The order of a non-abelian group is $pq$ such that $p<q$. Show that $p\mid q-1$ (without Sylow's theorem)
- Showing $x^{5}-ax-1\in\mathbb{Z}$ is irreducible
- How to find presentation of a group using GAP?
- If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
- Proving that $\left(\mathbb Q:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
- Cosemisimple Hopf algebra and Krull-Schmidt

I’m afraid that your question is very broad, and I doubt that it is answerable in full.

Let’s start here.

For $n\in \mathbb{N}$, denote by $\pi(n)$ the set of prime divisors of $n$.

Definition.Theprime graphof a finite group $G$, denoted $\Gamma(G)$, is a graph with vertex set $\pi(|G|)$ with an edge between primes $p$ and $q$ if and only if there is an element of order $pq$ in $G$.

Your question requires, among other things, knowing how many elements of order $pq$ there are in the group for every two primes $p,q$ dividing $|G|$, and in particular whether that number is nonzero. So answering your question would also tell us the answer to, “What are all the possible prime graphs of a finite group?” (In fact, it would be but a small corollary.) However, prime graphs are the subject of ongoing research, and there are still many unsolved problems in the world of prime graphs (most of which are easier than a full characterization).

We can perhaps reformulate your question into this language, though. Graphs can be generalized to hypergraphs, which are comprised of a vertex set $V$ and an edge set $E\subseteq \mathcal{P}(V)$ (i.e. the restriction that edges must connect at most two vertices in $V$ is removed).

For $n\in \mathbb{N}$, denote by $\overline{\pi}(n)$ the set of prime power divisors of $n$.

Definition.Define theweighted prime power hypergraphof a finite group $G$ as the hypergraph $\Gamma_H(G)$ with vertex set $\overline{\pi}(n)$ with a hyperedge $S$ if and only if the elements of $S$ are pairwise coprime and there exists an element of order $\prod_{s\in S}s$ in $G$. Furthermore, let each edge $S$ be given the weight $w_S$, which is equal to the number of elements of order $\prod_{s\in S}s$ in $G$.

Note that $\sum_{S\in E}w_S=|G|$. Naturally we can additionally define a strict total order on the edgeset of $\Gamma_H(G)$ where $S<T\Leftrightarrow \prod_{s\in S}s<\prod_{t\in T}t$. Sorting the weights by this order is equivalent to your $x_k$ sequence.

Thus your question is precisely “what are all possible ordered weight sequences of the prime power hypergraph of a finite group?” which is clearly a very complicated question.

The only thing I can say about really say about it is the following. For solvable groups $G$, it has been proven that $|\pi(|G|)|\leq 4\text{rank}(\Gamma_H(G))$ asymptotically. Additionally, it is conjectured that $|\pi(|G|)|\leq 3\operatorname{rank}(\Gamma_H(G))$ for all solvable groups. So, for solvable groups, you will always have to the count weights of edges whose size is up to at least one third the size of the vertex set.

To move forward, I would suggest you try to substantially narrow your question. You could restrict your question to a certain class of groups ($p$-groups? abelian groups? symmetric groups?). Furthermore I think it would be more likely to find an answer if you removed the ordering condition on $x_k$, as it adds a level of difficulty to the problem that I suspect outweighs any possible insight it would afford.

For further reading I’d recommend my answer to this similar question concerning sets of element orders, a related notion to yours.

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- Prove $\frac{a^3+b^3+c^3}{3}\frac{a^7+b^7+c^7}{7} = \left(\frac{a^5+b^5+c^5}{5}\right)^2$ if $a+b+c=0$
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