گروه خودریختی گراف توانی گروه های متناهی

گراف توانی یک گروه گرافی است با مجموعه راس های عناصر نابدیهی  گروه، و دو راس مجاور هستند اگر و تنها اگر یکی توانی از دیگری باشد. در این مقاله ابتدا روشی  کلی برای محاسبه گروه خودریختی گراف ها ارایه می دهیم. سپس با استفاده از این قضایا توصیفی برای گروه خودریختی های گراف توانی بدست می آوریم. همچنین  گروه خودریختی گراف  توانی گروه های آبلی و پوچ توان را برحسب سیلو زیرگروه های آنها محاسبه می کنیم. در انتها گروه خودریختی گراف توانی گروه های همودوری را به طور دقیق بدست می آوریم.

automorphism group of power graphs of finite groups

introductionimproving the quality, compliance rate and product life is a priority in manufacturing industries. if the performance of a process is evaluated by the lifetime of a product, it is clear that a larger lifetime indicates better product quality and higher process capability, and this process is accepted. for instance, the lifetime of electronic components exhibits a larger-the-better type of quality characteristic. the process capability index is one of the indicators for evaluating the capability of a process.  to measure the larger-the-better quality characteristic, a process capability index has been defined as ,  where   and  are the mean and standard deviation of the process, respectively,  and  is the lower specification limit.in various manufacturing and service processes, the assumption of normality is often not valid. the two-parameter exponential distribution has many applications in engineering, biological studies, epidemiology, and medical sciences. in engineering analysis, the location parameter is called the threshold value or warranty time and the scale parameter is the expected average as well as warranty life. if the lifetime of a manufactured product follows a two-parameter exponential distribution with location parameter  and scale parameter , then the capability index becomes . material and methodssince the performance capability index has been used in industries, point estimation, construction of confidence intervals, and testing the hypothesis about this index have been considered. however, the classical inference cannot be applied to .  therefore, we utilized the concepts of generalized pivotal quantity and generalized test variable for inference on this parameter. here, we suppose that different kinds of schemes such as double type ii censoring, progressive censoring, record values, and k-record values. in each case, the maximum likelihood and uniformly minimum variance unbiased estimators are derived. then, a generalized confidence interval and a generalized p-value are suggested.  the generalized confidence interval is evaluated by monte carlo simulation, and the presented approaches are illustrated using some real examples. results and discussionwe study the coverage probability and expected length of the proposed generalized confidence interval for  under a progressive censoring scheme. we found that the coverage probability is close to the confidence coefficient. also, it does not depend on the values of parameters and sample size. besides, the expected length decreases when  decreases (or the sample size increases).  the shortest expected length of the generalized confidence interval is when the value of  is close to the location parameter .conclusionthe following conclusions were drawn from this research.the minimum variance unbiased estimator for  has a closed-form under all considered schemes.the proposed generalized inference for  is a satisfactory and easy approach.this article contains all results in other papers such as lee et al. (2011) and gunasekera, and wijekularathna (2019).different indicators have been introduced to evaluate the process’s capability. some of the most important ones that are widely used in manufacturing industries have been introduced by wooluru et al. (2014). the methods introduced in this paper, can be generalized and used for these process capability indicators.