For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$ -subgroup of $G$ . We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of  $G$ .

Let $G$ be a $p$ -group and let $\unicode[STIX]{x1D712}$ be an irreducible character of  $G$ . The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$ . If $G$ is a maximal class $p$ -group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of  $p$ . If $|G|=p^{n}$ and $G$ has consecutive $p$ -power codegrees up to  $p^{n-1}$ , then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$ . We say that a $p$ -Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$ . When $G$ is $p$ -solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of $G$ .

Let $G$ be a finite group and let $p$ be a prime factor of $|G|$ . Suppose that $G$ is solvable and $P$ is a Sylow $p$ -subgroup of $G$ . In this note, we prove that $P{\vartriangleleft}G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all irreducible monomial $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of $G$ .

Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

Let $G$ be a finite solvable group and let $p$ be a prime. We prove that the intersection of the kernels of irreducible monomial $p$ -Brauer characters of $G$ with degrees divisible by $p$ is $p$ -closed.

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$ -Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$ -subgroup.

When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

We define several graphs related to the p-blocks of a solvable group. We bound the diameter of these graphs when the defect group associated with the block is either abelian or normal and when the group has odd order. We give examples to show that these bounds are met.

The prime vertex graph, $\Delta \left( X \right)$ , and the common divisor graph, $\Gamma \left( X \right)$ , are two graphs that have been defined on a set of positive integers $X$ . Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

Synthetic κ-opioid receptor (KOR) agonists induce dysphoric and pro-depressive effects and variations in the KOR (OPRK1) and prodynorphin (PDYN) genes have been shown to be associated with alcohol dependence. We genotyped 23 single nucleotide polymorphisms (SNPs) in the PDYN and OPRK1 genes in 816 alcohol-dependent subjects and investigated their association with: (1) negative craving measured by a subscale of the Inventory of Drug Taking Situations; (2) a self-reported history of depression; (3) the intensity of depressive symptoms measured by the Beck Depression Inventory-II. In addition, 13 of the 23 PDYN and OPRK1 SNPs, which were previously genotyped in a set of 1248 controls, were used to evaluate association with alcohol dependence. SNP and haplotype tests of association were performed. Analysis of a haplotype spanning the PDYN gene (rs6045784, rs910080, rs2235751, rs2281285) revealed significant association with alcohol dependence (p = 0.00079) and with negative craving (p = 0.0499). A candidate haplotype containing the PDYN rs2281285-rs1997794 SNPs that was previously associated with alcohol dependence was also associated with negative craving (p = 0.024) and alcohol dependence (p = 0.0008) in this study. A trend for association between depression severity and PDYN variation was detected. No associations of OPRK1 gene variation with alcohol dependence or other studied phenotypes were found. These findings support the hypothesis that sequence variation in the PDYN gene contributes to both alcohol dependence and the induction of negative craving in alcohol-dependent subjects.

Tactical emergency medical services (TEMS) bring immediate medical support to the inner perimeter of special weapons and tactics team activations. While initially envisioned as a role for an individual dually trained as a police officer and paramedic, TEMS is increasingly undertaken by physicians and paramedics who are not police officers. This report explores the ethical underpinnings of embedding a surgeon within a military or civilian tactical team with regard to identity, ethically acceptable actions, triage, responsibility set, training, certification, and potential future refinements of the role of the tactical police surgeon.

KaplanLJ, SiegelMD, EastmanAL, FlynnLM, RosenbaumSH, ConeDC, BlakeDP, MulhernJ. Ethical Considerations in Embedding a Surgeon in a Military or Civilian Tactical Team. Prehosp Disaster Med. 2012;27(6):1-6.