****************************************************** PR # 11162 We propose ASCA observations of X-ray emitting hot gas in Galaxy Groups, which contain about several member galaxies. The class has a position linking between the two levels of Claster of Galaxies and Elliptical Galaxy, both of which have X-ray emitting hot gas. Our aimes are as follows:(i)We examin if hot gas surrounding Galaxy Group exists and measure the ratio of gas mass to stellar mass in the sys- tem. (ii) Does the dark matter exist in Galaxy Groupand what is the distribution of it ? Also, we study dynamical state of GalaxyGroup. (iii) The measurements of metal abundance and metal mass in the hot gas in Galaxy Group are made. With them, we study thermal evolutions of the gas seen in the levels of Elliptical galaxy - Galaxy Group - Cluster of Galaxies. ******************************************************