Gary Walls

Problems

College Mathematics Journal, May 1, 1986

Multiplicative Lie algebras

Turkish Journal of Mathematics, Nov 22, 2019

A multiplicative Lie algebra is a group together with a "bracket function" that satisfies the bas... more

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Automorphism Groups

American Mathematical Monthly, Jun 1, 1986

Subgroup Perfect Codes in Cayley Graphs

SIAM Journal on Discrete Mathematics, 2020

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is... more Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is an independent set of Γ and every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset C of a group G is called a perfect code of G if there exists a Cayley graph of G which admits C as a perfect code. A group G is said to be code-perfect if every proper subgroup of G is a perfect code of G. In this paper we prove that a group is code-perfect if and only if it has no elements of order 4. We also prove that a proper subgroup H of an abelian group G is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of the Sylow 2-subgroup of G. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.

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The Real Orthogonal Group and Vahlen Matrices

Science Diliman, 2016

We prove the differentiability of a group homomorphism € ξ : O(4) →PSp(2) from the real orthogona... more We prove the differentiability of a group homomorphism € ξ : O(4) →PSp(2) from the real orthogonal group into the projective symplectic group where € ξ was constructed in (Canlubo and Reyes 2012). We describe higher dimensional analogues of € ξ. For € n ≥ 2, we consider a stereographic projection € Π : S n −1 →ˆ R n −1 from the unit sphere onto € ˆ R n −1 = R n −1 ∪{∞}. Applying Möbius transformations and , we embed the real orthogonal group into a projective subgroup of Vahlen matrices where is the Clifford group of the Clifford algebra of dimension .

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Finite groups with star-free noncyclic graphs

arXiv (Cornell University), Apr 23, 2016

For a finite noncyclic group G, let Cyc(G) be a set of elements a of G such that a, b is cyclic f... more For a finite noncyclic group G, let Cyc(G) be a set of elements a of G such that a, b is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G \ Cyc(G), having an edge between two distinct vertices x and y if x, y is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are K 1,n-free, where 3 ≤ n ≤ 6.

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Subgroup perfect codes in Cayley graphs

arXiv (Cornell University), Apr 3, 2019

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is... more Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is an independent set of Γ and every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset C of a group G is called a perfect code of G if there exists a Cayley graph of G which admits C as a perfect code. A group G is said to be code-perfect if every proper subgroup of G is a perfect code of G. In this paper we prove that a group is code-perfect if and only if it has no elements of order 4. We also prove that a proper subgroup H of an abelian group G is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of the Sylow 2-subgroup of G. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.

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On subgroup perfect codes in Cayley graphs

arXiv (Cornell University), Jun 19, 2020

A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjace... more A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \ C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing the identity element such that (A, H) is a tiling of G in the sense that every element of G can be uniquely expressed as the product of an element of A and an element of H. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient conditions for a subgroup of a finite group to be a subgroup perfect code, a few results involving 2-subgroups in the study of subgroup perfect codes, and several results on subgroup perfect codes of metabelian groups, generalized dihedral groups, nilpotent groups and 2-groups.

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Forcing a Finite Group to Be Abelian

Mathematical Proceedings of the Royal Irish Academy, 2011

H. Zassenhaus proved that a group G is abelian provided N G (A) = C G (A) for all abelian subgrou... more H. Zassenhaus proved that a group G is abelian provided N G (A) = C G (A) for all abelian subgroups A of G. In this paper we seek to extend this result by proving that a finite group is abelian if and only if N G (C G (x)) = C G (C G (x)) for all nontrivial elements x ∈ G. This result is a consequence of the fact that a finite, nonabelian group must contain some x ∈ Z(G) such that C G (C G (x)) is properly contained in N G (C G (x)). Other "different" conditions are investigated that imply that a finite group must be abelian.

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Sums of 2-by-2 Matrices with Determinant One

Problems

The College Mathematics Journal, 1986

On the Nilpotent Residual Norm of a Group and the Structure of S-Groups

by Gary Walls and zhencai Shen

Mediterranean Journal of Mathematics, 2022

Abstract. For a finite group G, we define the subgroup S(G) to be the intersection of the normal... more Abstract. For a finite group G, we define the subgroup S(G) to be the
intersection of the normalizers of the nilpotent residuals of all subgroups
of G. Set S0 = 1 and define Si+1(G)/Si(G) = S(G/Si(G)) for i ≥ 1.
The terminal term of this upper series is denoted by S∞(G). This upper
series implies a lot of information on the structure of G. In this paper,
we solve several basic problems on S(G). If G = S(G), we call the finite
group G an S-group. The new class of S-groups are investigated and
some open problems on S-groups are posed. Furthermore, we develop
the research on S∞(G) by a new idea and unify some known results.
Mathematics Subject Classification. Primary 20D10; Secondary 20D15.
Keywords. Finite group, nilpotent residual, S-group, nilpotence class.

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Problems

The College Mathematics Journal, 1986

Power graphs of (non)orientable genus two

The power graph Γ_G of a finite group G is the graph whose vertex set is the group, two distinct ... more

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On subgroup perfect codes in Cayley graphs

European Journal of Combinatorics, 2021

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is... more Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is an independent set of Γ and every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset C of a group G is called a perfect code of G if there exists a Cayley graph of G which admits C as a perfect code. A group G is said to be code-perfect if every proper subgroup of G is a perfect code of G. In this paper we prove that a group is code-perfect if and only if it has no elements of order 4. We also prove that a proper subgroup H of an abelian group G is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of the Sylow 2-subgroup of G. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.

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Power graphs of (non)orientable genus two

Communications in Algebra, 2019

The power graph Γ G of a finite group G is the graph whose vertex set is the group, two distinct ... more

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Finite Groups with metacyclic QTI-subgroups

Journal of Algebra and Its Applications, 2016

Let G be a finite group. A subgroup A of G is called a TI-subgroup of G if A ∩ Ax = 1 or A for al... more Let G be a finite group. A subgroup A of G is called a TI-subgroup of G if A ∩ Ax = 1 or A for all x ∈ G. A subgroup H of G is called a QTI-subgroup if CG(x) ⊆ NG(H) for every 1 ≠ x ∈ H, and a group G is called an MCTI-group if all its metacyclic subgroups are QTI-subgroups. In this paper, we show that every nilpotent MCTI-group is either a Dedekind group or a p-group and we completely classify all the MCTI-p-groups. We show that all MCTI-groups are solvable and that every nonnilpotent MCTI-group must be a Frobenius group having abelian kernel and cyclic complement.

3 RINGS ASSOCIATED TO COVERINGS OF FINITE p-GROUPS

There is a natural way to associate a ring to any group. In this paper we characterize the rings ... more There is a natural way to associate a ring to any group. In this paper we characterize the rings associated to finite p-groups when the covering consists of maximal cyclic subgroups and subgroups of order p 2. 2010 Mathematics Subject Classification. 16S60. Key words and phrases. finite p-groups, covers of groups, rings of functions. Research of the second author supported by the Louisiana BoR [LEQSF(2012-15)-RD-A-20].

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Some Sufficient conditions for a group to be abelian

TURKISH JOURNAL OF MATHEMATICS

A group is said to satisfy a word w in the symbols {x, x −1 , y, y −1 } provided that if the 'x' ... more A group is said to satisfy a word w in the symbols {x, x −1 , y, y −1 } provided that if the 'x' and 'y' are replaced by arbitrary elements of the group then the equation w = 1 is satisfied. This paper studies certain equations in words, as above, which together with other conditions imply that groups which satisfy these equations and conditions must be abelian.

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Finite groups with star-free noncyclic graphs

Open Mathematics

For a finite noncyclic group G, let Cyc(G) be the set of elements a of G such that 〈a, b〉 is cycl... more For a finite noncyclic group G, let Cyc(G) be the set of elements a of G such that 〈a, b〉 is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G ∖ Cyc(G), having an edge between two distinct vertices x and y if 〈x, y〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are K1,n-free, where K1,n is a star and 3 ≤ n ≤ 6.

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