Equations solvable by radicals in a uniquely divisible group

Equations Solvable by Radicals in a Uniquely Divisible Group

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B ∈ G as fixed coefficients, and X ∈ G as the unknown. Certain word equations, such as XAXAX = B, have solutions in terms of radicals: X = A(ABA)A while others such as XAX = B do n...

Word equations in a uniquely divisible group

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B ∈ G as fixed coefficients, and X ∈ G as the unknown. Certain word equations, such as XAXAX = B, have solutions in terms of radicals: X = A−1/2(A1/2BA1/2)1/3A−1/2, while others su...

Uniquely Solvable Puzzles and Fast Matrix Multiplication

In 2003 Cohn and Umans introduced a new group-theoretic framework for doing fast matrix multiplications, with several conjectures that would imply the matrix multiplication exponent ω is 2. Their methods have been used to match one of the fastest known algorithms by Coppersmith and Winograd, which runs in O(n2.376) time and implies that ω ≤ 2.376. This thesis discusses the framework that Cohn a...

Solvable $L$-subgroup of an $L$-group

In this paper, we study the notion of solvable $L$-subgroup of an $L$-group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an $L$-subgroup of an $L$-group in the sense of Wu rather than Liu.

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup