13
votes
0answers
460 views

What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11, London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...
7
votes
0answers
176 views

A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting? Proposition There exists an absolute constant $c$ such ...
6
votes
1answer
351 views

Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$. Question: For what groups does there exist a Hamiltonian ...
1
vote
1answer
118 views

Max order for which connected Cayley Graphs are known to be Hamiltonian

There is a well-known conjecture that all connected Cayley graphs are Hamiltonian. For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?
25
votes
2answers
880 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 ...
14
votes
1answer
550 views

Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function? Background: The counting function, $f(n)$, is a ...
6
votes
2answers
460 views

Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?
2
votes
2answers
329 views

Automorphism group action leads to a “quotient graph”

Let $G$ be a simple (finite) graph. Consider the next natural equivalence relation $\sim$ on $V(G)$: $u\sim v$ iff there exists and automorphism $\phi\in Aut(G)$, such that $\phi(u)=v$. Define a new ...
5
votes
1answer
165 views

How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?

Let $G \leq {\rm S}_n$ be a finite permutation group, and let $S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed under inversion and which does not contain the identity. The growth ...
5
votes
2answers
602 views

Orthogonal orthomorphisms of order 2

EDIT: There is an additional requirement that the composition of the orthomorphisms will also be order 2 (see my answer below). A full proof is not needed, I will be happy with any argument which ...
14
votes
0answers
233 views

Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction. Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...
12
votes
1answer
252 views

Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$

Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...
0
votes
0answers
28 views

Minimum number of solutions in a system of equalities and non-equalities

Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$. Find the minimum number of solution of the system $$P_{2i} + P_{2i+1} = \lambda_i, ...
15
votes
1answer
563 views

Lower bounds on the number of elements in Sylow subgroups

I posted this question on Math.SE (link), but it didn't get any answers so I'm going to ask here. This is an edited version of the question. Let $p$ be a prime and $n \geq 1$ some integer. ...
18
votes
1answer
1k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order ...
7
votes
2answers
524 views

Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$. There's a ...
3
votes
2answers
344 views

The cycle structure of twisted wires, connected

Suppose you have $n$ (blue) wires linearly arrayed at junction box $A$, connected to a remote junction box $B$, where the wires are now arrayed along a line in a randomly permuted order, i.e., each ...
12
votes
4answers
1k views

Has any attempt been made to classify finite groupoids?

I recently stumbled upon the Mathieu groupoid and I found them fascinating. It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
0
votes
2answers
447 views

primes dividing binomial coefficients

Dear All, I am considering maximal subgroups of odd index in Alternating and Symmetric groups, and this leeds me to some questions on binomial coefficients that I presently do not know and that I ...
8
votes
1answer
353 views

When Cayley graphs of the symmetric group wrt generating sets of transpositions are isomorphic?

Dear All, I thought the following question might be well-known, but couldn't find anywhere, so decided to ask here: Let $A$ and $B$ be two generating sets for $S_n$, consisting of transpositions. ...
4
votes
1answer
262 views

From the chinese remainder theorem to products of transitive G-sets

Note: I am aware of the question Analog to the Chinese Remainder Theorem in groups other than Z_n. For an abelian group $A$, every transitive $A$-set $M$ is of course isomorphic, as an $A$-set, to a ...
5
votes
1answer
249 views

Group not leaving subset invariant

Let $Y,X$ be two sets of size n,m. Let $Y\subset X$. What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$? Here I mean that the only permutation which permutes elements of ...
17
votes
5answers
2k views

Generating finite simple groups with $2$ elements

Here is a very natural question: Q: Is it always possible to generate a finite simple group with only $2$ elements? In all the examples that I can think of the answer is yes. If the answer is ...
3
votes
1answer
385 views

Looking for deterministic criteria to generate the symmetric group?

So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its natural action on the set $T=\{1,2,\ldots,N\}$. Say that $H\leq S_N$ is a subgroup which acts ...
1
vote
1answer
276 views

Large subgroups of the Hamming cube

Let's consider the abelian group $\mathbb{Z}^N_2$ equipped with the Hamming metric (the hypercube). Suppose I have a subgroup of this hypercube (not necessarily a subcube) which is generated by a set ...
6
votes
0answers
484 views

When is Hom(G, H) the same size as Hom(H, G)?

Let $G$ and $H$ be finite groups. Consider the ratio $$r_{G, H} \equiv {|Hom(G, H)| \over{|Hom(H,G)|}}$$ My question is When is $r_{G, H} = 1$? Can we characterize the pairs of groups $(G, H)$ ...
7
votes
1answer
974 views

Burnside's Lemma and Geometry

I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action. I wonder if there is any ...
7
votes
2answers
714 views

Are vertex and edge-transitive graphs determined by their spectrum?

A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges. The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the ...
31
votes
1answer
2k views

orders of products of permutations

Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...
2
votes
1answer
304 views

Transitivity-related property of finite permutation groups

Let $\cal F$ denote the group of all finitely-supported permutations of $\mathbb N$. Say that a finite subgroup $G$ of $\cal F$ is singular if $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ but ...
7
votes
2answers
384 views

Elements living in the conjugacy class and in the centralizer of an m-cycle in Am

Let m>1 be an odd natural number, x a m-cycle in Am, the alternating group in m letters, C the conjugacy class of x in Am. Questiom: How can I describe the elements in the set { j | x^j in C} in ...
24
votes
2answers
874 views

Is $\varphi(n)/n$ the maximal portion of $n$-cycles in a degree $n$ group?

Let $G$ be a degree $n$ group, i.e., a subgroup of the symmetric group $S_n$. Let $p(G)$ be the number of $n$-cycles in $G$ divided by the size of $G$. Examples: If $G$ is a cyclic transitive ...
8
votes
2answers
1k views

Number of faithful representations of a finite group

Is it known how many faithful linear representations a finite group G has on a complex vector space of given dimension? What if G is abelian? I would even be interested in this special case: the ...
15
votes
3answers
990 views

Injective proof about sizes of conjugacy classes in S_n

It's not hard to count the number of permutations in a given conjugacy class of Sn. In particular, the number of permutations in Sn whose cycle decomposition has ci i-cycles is n!/(Πi=1n ci!ici). ...