A Robinson characterization of finite PσTP\sigma T-groups

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $\sigma$-full if $G$ has a Hall $\sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $\sigma$-permutable in $G$ provided $G$ is $\sigma$-full and $A$ permutes with all Hall $\sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $\sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${\sigma}$-permutable subgroup of $H$ and $H$ is a ${\sigma}$-permutable subgroup of $G$, then $K$ is a ${\sigma}$-permutable subgroup of $G$.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.0250

PRISMA flowchart.

BackgroundHIV testing services (HTS) are the first steps in reaching the UNAIDS 95-95-95 goals to achieve and maintain low HIV incidence. Evaluating the effectiveness of different demand creation interventions to increase uptake of efficient and effective HTS is useful to prioritize limited programmatic resources. This review was undertaken to inform World Health Organization (WHO) 2019 HIV testing guidelines and assessed the research question, “Which demand creation strategies are effective for enhancing uptake of HTS?” focused on populations globally.Methods and findingsThe following electronic databases were searched through September 28, 2021: PubMed, PsycInfo, Cochrane CENTRAL, CINAHL Complete, Web of Science Core Collection, EMBASE, and Global Health Database; we searched IAS and AIDS conferences. We systematically searched for randomized controlled trials (RCTs) that compared any demand creation intervention (incentives, mobilization, counseling, tailoring, and digital interventions) to either a control or other demand creation intervention and reported HTS uptake. We pooled trials to evaluate categories of demand creation interventions using random-effects models for meta-analysis and assessed study quality with Cochrane’s risk of bias 1 tool. This study was funded by the WHO and registered in Prospero with ID CRD42022296947.We screened 10,583 records and 507 conference abstracts, reviewed 952 full texts, and included 124 RCTs for data extraction. The majority of studies were from the African (N = 53) and Americas (N = 54) regions. We found that mobilization (relative risk [RR]: 2.01, 95% confidence interval [CI]: [1.30, 3.09], p p N = 4 RCTs), couple-oriented counseling (RR: 1.98, 95% CI [1.02, 3.86], p p N = 4 RCTs), peer-led interventions (RR: 1.57, 95% CI [1.15, 2.15], p p N = 10 RCTs), motivation-oriented counseling (RR: 1.53, 95% CI [1.07, 2.20], p p N = 4 RCTs), short message service (SMS) (RR: 1.53, 95% CI [1.09, 2.16], p p N = 5 RCTs), and conditional fixed value incentives (RR: 1.52, 95% CI [1.21, 1.91], p p N = 11 RCTs) all significantly and importantly (≥50% relative increase) increased HTS uptake and had medium risk of bias.Lottery-based incentives and audio-based interventions less importantly (25% to 49% increase) but not significantly increased HTS uptake (medium risk of bias). Personal invitation letters and personalized message content significantly but not importantly (ConclusionsMobilization, couple- and motivation-oriented counseling, peer-led interventions, conditional fixed value incentives, and SMS are high-impact demand creation interventions and should be prioritized for programmatic consideration. Reduced duration counseling and video-based interventions are an efficient and effective alternative to address staffing shortages. Investment in demand creation activities should prioritize those with undiagnosed HIV or ongoing HIV exposure. Selection of demand creation interventions must consider risks and benefits, context-specific factors, feasibility and sustainability, country ownership, and universal health coverage across disease areas.</div

Summary of effect sizes of pooled estimates of uptake from meta-analysis.

Pooled RR effect sizes ranged from near 1.0 (indicating no improvement in uptake of HTS) through nearly 7.0 (indicating a large effect size); not important improvements (RR p p > 0.05). CI, confidence interval; HTS, HIV testing service; RR, relative risk; SMS, short message service; vs., versus.</p

(A and B) Risk of bias summary.

Panel (A): Pooled studies. Studies are grouped by the meta-analysis that they contributed to. Panel (B): Not pooled studies. Studies that were not pooled are listed in alphabetical order.</p

Prisma checklist.

BackgroundHIV testing services (HTS) are the first steps in reaching the UNAIDS 95-95-95 goals to achieve and maintain low HIV incidence. Evaluating the effectiveness of different demand creation interventions to increase uptake of efficient and effective HTS is useful to prioritize limited programmatic resources. This review was undertaken to inform World Health Organization (WHO) 2019 HIV testing guidelines and assessed the research question, “Which demand creation strategies are effective for enhancing uptake of HTS?” focused on populations globally.Methods and findingsThe following electronic databases were searched through September 28, 2021: PubMed, PsycInfo, Cochrane CENTRAL, CINAHL Complete, Web of Science Core Collection, EMBASE, and Global Health Database; we searched IAS and AIDS conferences. We systematically searched for randomized controlled trials (RCTs) that compared any demand creation intervention (incentives, mobilization, counseling, tailoring, and digital interventions) to either a control or other demand creation intervention and reported HTS uptake. We pooled trials to evaluate categories of demand creation interventions using random-effects models for meta-analysis and assessed study quality with Cochrane’s risk of bias 1 tool. This study was funded by the WHO and registered in Prospero with ID CRD42022296947.We screened 10,583 records and 507 conference abstracts, reviewed 952 full texts, and included 124 RCTs for data extraction. The majority of studies were from the African (N = 53) and Americas (N = 54) regions. We found that mobilization (relative risk [RR]: 2.01, 95% confidence interval [CI]: [1.30, 3.09], p p N = 4 RCTs), couple-oriented counseling (RR: 1.98, 95% CI [1.02, 3.86], p p N = 4 RCTs), peer-led interventions (RR: 1.57, 95% CI [1.15, 2.15], p p N = 10 RCTs), motivation-oriented counseling (RR: 1.53, 95% CI [1.07, 2.20], p p N = 4 RCTs), short message service (SMS) (RR: 1.53, 95% CI [1.09, 2.16], p p N = 5 RCTs), and conditional fixed value incentives (RR: 1.52, 95% CI [1.21, 1.91], p p N = 11 RCTs) all significantly and importantly (≥50% relative increase) increased HTS uptake and had medium risk of bias.Lottery-based incentives and audio-based interventions less importantly (25% to 49% increase) but not significantly increased HTS uptake (medium risk of bias). Personal invitation letters and personalized message content significantly but not importantly (ConclusionsMobilization, couple- and motivation-oriented counseling, peer-led interventions, conditional fixed value incentives, and SMS are high-impact demand creation interventions and should be prioritized for programmatic consideration. Reduced duration counseling and video-based interventions are an efficient and effective alternative to address staffing shortages. Investment in demand creation activities should prioritize those with undiagnosed HIV or ongoing HIV exposure. Selection of demand creation interventions must consider risks and benefits, context-specific factors, feasibility and sustainability, country ownership, and universal health coverage across disease areas.</div

Geographic distribution of included trials.

Key and bar chart identify the total number of trials included from each country on the map. The rworldmap [cran.r-project.org] package in R was used to obtain the publicly available map (South A (2011). “rworldmap: A New R package for Mapping Global Data.” The R Journal, 3(1), 35–43. ISSN 2073-4859); the base layer map file can be found: https://code.google.com/archive/p/rworld/source/default/source. (DOCX)</p

Funnel plots and Egger’s tests to assess publication bias.

Funnel plots and Egger’s tests to assess publication bias.</p

Meta-analysis plots of HTS uptake in relative and absolute differences.

Panels (A–O) 1: INCENTIVES: (A) conditional fixed value incentives, (B) lottery-based incentives, 2: MOBILIZATION: (C) mobilization, 3: TAILORED or TARGETED: (D) peer-led interventions, (E) personalized messages, (F) personal invitation letters, 4: MESSAGES & COUNSELING: (G) HIV-specific information and counseling, (H) HIV-specific information with economic empowerment, (I) couples counseling, (J) motivation-oriented counseling, (K) reduced duration counseling, 5: DIGITIZATION: (L) video-based vs text, (M) video-based vs. in-person, (N) audio information, (O) SMS. CI, confidence interval; CRCT, cluster-randomized trial; RCT, randomized controlled trial; RD: risk difference; REML, restricted maximum likelihood; RR, relative risk; SMS, short message service; SOC, standard of care.</p

Subgroup analyses for meta-analyses with high statistical heterogeneity.

Subgroup analyses for meta-analyses with high statistical heterogeneity.</p

Trim-and-fill adjusted estimates of uptake and yield.

Trim-and-fill adjusted estimates of uptake and yield.</p