Abstract We add a missed hypothesis in the statement of Theorem A(b) “Conjugacy classes and union cosets normal subgroups", Monatsh. Math. 201 , No 2, 349–358 (2023), give corrected proof.

Todd and Coxeter's method for enumerating cosets of nitely generated subgroups in nitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since preex string rewriting is also an appropriate method to study this problem, preex string rewriting methods have been compared to Tc. We recall and compare two of them briee...

In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group SO − (2n, q). Here q is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formul...

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemeredi regularity lemma in graph theory. It shows that for any abelian group G and any bounded function f : G → [0, 1], there exists a subgroup H ≤ G of bounded index such that, when restricted to most cosets of H, the function f is pseudorandom in the sense that all its nontrivial Fourier coefficients are ...

We give an expository account of our computational proof that every position of Rubik’s Cube can be solved in 20 moves or less, where a move is defined as any twist of any face. The roughly 4.3 × 1019 positions are partitioned into about two billion cosets of a specially chosen subgroup, and the count of cosets required to be treated is reduced by considering symmetry. The reduced space is sear...

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U1, U2, · · · , Um} be the set of cosets of U in G. We say a matrix H = [hij ] order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if ∑ 1≤t≤k hith −1 jt = λij1U1 + · · · + λijmUm (∃λij1, · · · , ∃λijm ∈ Z) for any i ̸= j. On the other hand, in our previous article we def...

The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography. In particular, it is one of the most prominent candidates for secure post-quantum cryptography. Understanding its quantum complexity is therefore an important goal. We show that under quantum polynomial time reductions, LWE is equivalent to a relaxed version of the dihedral coset...

Let p be a prime Zp the eld of residue classes mod p and Up the cyclic group of automorphisms of the additive group Zp We will identify Up with the non zero elements of Zp For any positive integer n dividing p there is exactly one subgroup C of Up of index n given by C fx j x Zp x g if jCj k then p kn The orbits of C on Zp are f g C together with the cosets of C in Up If y is a primitive root i...

This paper introduces and develops a general framework for studying triple factorisations of the form G = ABA of finite groups G, with A and B subgroups of G. We call such a factorisation nondegenerate if G 6= AB. Consideration of the action of G by right multiplication on the right cosets of B leads to a nontrivial upper bound for |G| by applying results about subsets of restricted movement. F...

Abstract. This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let a1G1, . . . , akGk be left cosets in a group G such that {aiGi} k i=1 covers each element of G at least m times but none of its proper subsystems does. We show that if G is cyclic, or G is finite and G1, . . . , Gk are normal Hall subgroups of G, then the inequality k > m + f([G : ⋂ k i=...