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0answers
63 views

Relations between Arboreal Group Theory and Tree Group Actions? [migrated]

By a tree group action, we mean an action of a group $G$ over the infinite regular binary tree $T_2$ such that for each $g \in G$, the mapping $x \rightarrow g.x$ is an automorphism of $T_2$; these ...
2
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0answers
50 views

The Alexander-Conway polynomial: from knots to braids? [migrated]

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
3
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2answers
111 views

TCS oriented refs/survey on group theoretic word problem

The word problem for groups was shown to be Turing-complete in 1955 but has many decidable subcases. This problem arose more in mathematical group theory than in theoretical computer science, but now ...
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votes
1answer
150 views

Some questions (still) unresolved about braids?

I was looking for interesting questions pertaining to braid theory. I don't know if the following are considered important, but I'd like to ask: (1) in relation with the following link, is it true ...
0
votes
1answer
166 views

Classifying noetherian simple groups by order type?

A (possibly infinite) group $G$ is noetherian if it satisfies the following equivalent conditions: (1) every subgroup of $G$ is finitely generated, (2) there is no infinite strict ascending chain of ...
2
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0answers
92 views

Extending the notion of independence

Background I was looking for a formulation of 'free sets' and 'independent sets' from linear algebra that would extend to groups. This question was considered here but I couldn't find a satisfactory ...
3
votes
3answers
110 views

On proving it is hard to compute $g^{rb}$ with knowledge of $r$, given $g, g^a, g^{ab}$

I am trying to prove the following Given $g, g^a, g^{ab}$ it is hard to compute $r, g^r, g^{rb}$, for some arbitrarily chosen value of $r$ where $g ∈ \mathbb{G}, \mathbb{G}$ is a cyclic group ...
9
votes
1answer
72 views

Encoding sets of permutations with a generating set and a set of excluded elements

Polynomial-time algorithms are known for finding generating sets of permutation groups, which is interesting since we can then represent those groups succinctly without giving up on polynomial-time ...
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0answers
52 views

Upper bound on the number of vertex-transitive graphs

Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
0
votes
1answer
90 views

How to go about finding the most “complex” function?

Intro Hi, I'm a hobbyist, with no formal education, tinkering with SAT solving and boolean algebra minimization. So expect bad terminology. I hope you will forgive me I'm asking a wrong question in a ...
12
votes
0answers
198 views

Are there other proofs for Barrington's theorem?

I know that you can use other non-solvable groups, but is there a proof that uses a completely different approach? In case someone would not know the theorem: ...
8
votes
0answers
127 views

Difficulty in understanding the quantum algorithm for the abelian hidden subgroup problem

I've difficulty in understanding the last steps of the AHSP algorithm. Let $G$ be an abelian group and $f$ be the function which hides the subgroup $H$. Let $G^*$ represent the dual group of $G$. ...
8
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0answers
128 views

Complexity of checking if AB intersects C

Let $A,B,C$ be subsets of a nonabelian group $G$, and assume we know the structure of $G$ "fairly well" (e.g., $G = S_n$ or $A_n$). Assume that group operations take $O(1)$ time. Is it ...
7
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0answers
153 views

Is there a candidate for a post-quantum one-way group action?

Is there a known family of group actions with a designated element in the set that is being acted on, where it is known how to efficiently $\:$ sample (essentially uniformly) from the groups, ...
1
vote
2answers
218 views

Primitive Recursive Isomorphisms

What is the relationship between invertible primitive recursive functions (that is, a primitive recursive function that is an isomorphism) and all primitive recursive functions? Can every primitive ...
14
votes
1answer
297 views

Complexity of recognizing vertex-transitive graphs

I am not knowledgeable in the area of complexity theory involving groups so I apologize if this is a well known result. Question 1. Let $G$ be a simple undirected graph of order $n$. What is the ...
30
votes
12answers
1k views

Applications of representation theory of the symmetric group

Inspired by this question and in particular the final paragraph of Or's answer, I have the following question: Do you know of any applications of the representation theory of the symmetric group ...
12
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2answers
537 views

Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to ...
3
votes
1answer
105 views

Choices for the group in Public Key Cryptography

I am only aware of two alternatives for the group used in Diffie-Hellman scheme (and similar ones) where logarithms are conjectured to be hard. Those are $\mathbb{F}_p$ and Elliptic Curves. Are there ...
6
votes
1answer
212 views

What is the most efficient algorithm for deciding if an element is the least in its orbit?

Given a group $G$ acting on a set $X$ with a total order $\leq$ and an $x\in X$, what is the most efficient algorithm for deciding whether or not x is the least element in its orbit, in other words, ...
12
votes
2answers
440 views

Complexity of Membership-Testing for finite abelian groups

Consider the following abelian-subgroup membership-testing problem. Inputs: A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with ...
11
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0answers
141 views

Do there exist groups with word problems in arbitrary P-degrees?

It has been known for a long time that, given any r.e. Turing degree, there is a finitely presented group whose word problem is in that degree. My question is whether the same thing is true for ...