Questions tagged [gr.group-theory]

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Relation between automorphism group of a linear code and its dual code

Are there any strong connections between automorphism groups of codes that are dual codes of each other? I am looking for statements like one charcterizes other or one gives bounds on other etc. In ...
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Generating set of a group and relation to diameter?

Is there an efficient algorithm to test if a given subset of symmetric group generates the symmetric group? Drawing the Cayley graph and testing for all pairs reachability is one way however that ...
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What is the hardest instance for the group isomorphism problem?

Two groups $(G,\cdot)$ and $(H, \times)$ are said to be isomorphic iff there exists a homomorphism from $G$ to $H$ which is bijective. The group isomorphism problem is as follows: given two groups, ...
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What is the probability that a random Boolean function has a trivial automorphism group?

Given a Boolean function $f$, we have the automorphism group $Aut(f) = \{\sigma \in S_n\ \mid \forall x, f(\sigma(x)) = f(x) \}$. Are there any known bounds on $Pr_f(Aut(f) \neq 1)$? Is there ...
119 views

Deterministic context-free languages that can be represented as the word problem of a group

Consider a group $G$. We call $G$ virtually free is it contains a free subgroup of finite index. If $G$ is finitely generated by some set $X \subseteq G$ one can consider the word problem $WP(G)$ ...
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Complexity of the coset intersection problem

Given the symmetry group $S_n$ and two subgroups $G, H\leq S_n$, and $\pi\in S_n$, does $G\pi\cap H=\emptyset$ hold? As far as I know, the problem is known as the coset intersection problem. I am ...
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Explicit error bounds on the abelian hidden subgroup problem

What are some explicit forms for the error probability in the typical quantum abelian hidden subgroup algorithm as a function of oracles queries? Ettinger, Hoyer, and Knill give a result that the ...
271 views

$\ell_1$ norm of Fourier coefficients vector for the hypercube

Let $G$ be the normzlied hypercube graph on $2^d$. It is a Cayley graph and it is well known that its eigenvalues are given by $\lambda_r = 1-2\frac{|r|}{d}$ for every $r \in \{0,1\}^d$. Given a ...
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Recent progress in permutation groups algorithms?

I am interested in algorithms for finite groups as implemented in the GAP package. It seems that all known algorithms in this field deal with permutation groups/matrix groups; two fundamental ones are ...
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Generalizations of the determinant/permanent problem?

A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with ...