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About Q1, this is standard. Take a strong generating set. List the strong generators in inverse order of the length of their stabiliser. I.e., first come the generators that fix all but the last point …

I just came back from a beach which features a large Rubik's cube (2m high). The base of the cube is not visible and the top is not coloured. The four vertical sides are each divided $3\times 3$ into …

The following came up in a problem on graph reconstruction. It isn't very important, but I thought some people here might find it interesting and not too trivial (I'm not a group theorist). Take a s …

Suppose we have a permutation $\pi$ on $1,2,\ldots,n$ and want to determine the parity (odd or even) of $\pi$. The standard method is find the cycles of $\pi$ and recall that the parity of $\pi$ equa …

Every finite simple group can be generated by two elements. Except in the case of prime order, one of the elements can have order 2. See here for example.

If the input has fully-expanded polynomials, then this is equivalent to graph isomorphism. In one direction, given a graph, create a variable for each vertex and consider the polynomial $\prod_{vw\in …

Every graph is an induced subgraph of some vertex transitive graph (a result I first learned from Chris). In fact Fink and Ruiz showed in 1984 that for any graph $H$, there is a circulant graph $G~$ …

NEW VERSION: (What was I thinking?) A greedy algorithm gives a stronger result. THEOREM. Consider any family $\mathcal F$ of $n$ 4-subsets of $\lbrace 1,\ldots,n\rbrace$. Then there is a set $X\sub …

If I understand my own 1979 catalogue of small transitive graphs, this happens first at 12 vertices. The simplest example to describe (L10 in the catalogue): take the tetrahedon and cut off each of t …

The number of ordered pairs of commuting functions is A181162. I agree with those counts up to n=7. There is little in OEIS that helps to answer the asymptotics question. Incidentally, the probabil …

There is no simple exact answer to this question. The harmonic mean of the automorphism group sizes is related to the ratio between the labelled and unlabelled graph counts (since the number of graph …

For $3\le d\le n-4$ the group size is almost always 1. The next most likely group size is 2, which most probably occurs due to a transposition (I don't know where this is proved formally). There is no …

Here's a partial answer. Take the generators to be a set of equal length cycles that are disjoint except that they have one point in common. For example $\langle (1,2,3,4), (1,5,6,7), (1,8,9,10)\ran …

Peter and YCor already gave a counterexample, so this answer is just some additional commentary. I'll ignore loops for simplicity. If $\varGamma$ is a permutation group on $\lbrace 1,\ldots, n \rbrace …

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 g …