Uni-width subgroups, universal elements, and lambda number of finite groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915747619405824 |
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| author | Sarkar, Siddhartha |
| author_facet | Sarkar, Siddhartha |
| contents | A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup denoted by $U(1;G)$. We then show that the prime factors of the order of $U(1;G)$ influence the structure decomposition of its Fitting subgroup ${\mathrm{Fit}}(G)$. A power graph $Γ_G$ of a finite group is defined by $G$ being its set of vertices, and a pair of distinct elements $x,y \in G$ are connected by an edge if either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. A universal element of a graph is a vertex that is adjacent to each of the remaining vertices. Our following result shows that a power graph $Γ_G$ of a finite non-trivial group admits a non-identity universal element if and only if it is either cyclic or a generalized quaternion $2$-group. The lambda number $λ(G)$ of a finite group $G$ is a measure of the least number of colors required for an $L(2,1)$-type of vertex coloring on $Γ_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $λ(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $λ(G) > |G|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09818 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Uni-width subgroups, universal elements, and lambda number of finite groups Sarkar, Siddhartha Group Theory Combinatorics Primary 05C25 Secondary 05C78, and 20D25 A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup denoted by $U(1;G)$. We then show that the prime factors of the order of $U(1;G)$ influence the structure decomposition of its Fitting subgroup ${\mathrm{Fit}}(G)$. A power graph $Γ_G$ of a finite group is defined by $G$ being its set of vertices, and a pair of distinct elements $x,y \in G$ are connected by an edge if either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. A universal element of a graph is a vertex that is adjacent to each of the remaining vertices. Our following result shows that a power graph $Γ_G$ of a finite non-trivial group admits a non-identity universal element if and only if it is either cyclic or a generalized quaternion $2$-group. The lambda number $λ(G)$ of a finite group $G$ is a measure of the least number of colors required for an $L(2,1)$-type of vertex coloring on $Γ_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $λ(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $λ(G) > |G|$. |
| title | Uni-width subgroups, universal elements, and lambda number of finite groups |
| topic | Group Theory Combinatorics Primary 05C25 Secondary 05C78, and 20D25 |
| url | https://arxiv.org/abs/2202.09818 |