We present the results of a search for extended X-ray sources and their corresponding galaxy groups from 800-ks Chandra coverage of the All-wavelength Extended Groth Strip International Survey (AEGIS). This yields one of the largest X-ray selected galaxy group catalogs from a blind survey to date. The red-sequence technique and spectroscopic redshifts allow us to identify 100$%$ of reliable sources, leading to a catalog of 52 galaxy groups. The groups span the redshift range $zsim0.066-1.544$ and virial mass range $M_{200}sim1.34times 10^{13}-1.33times 10^{14}M_odot$. For the 49 extended sources which lie within DEEP2 and DEEP3 Galaxy Redshift Survey coverage, we identify spectroscopic counterparts and determine velocity dispersions. We select member galaxies by applying different cuts along the line of sight or in projected spatial coordinates. A constant cut along the line of sight can cause a large scatter in scaling relations in low-mass or high-mass systems depending on the size of cut. A velocity dispersion based virial radius can more overestimate velocity dispersion in comparison to X-ray based virial radius for low mass systems. There is no significant difference between these two radial cuts for more massive systems. Independent of radial cut, overestimation of velocity dispersion can be created in case of existence of significant substructure and also compactness in X-ray emission which mostly occur in low mass systems. We also present a comparison between X-ray galaxy groups and optical galaxy groups detected using the Voronoi-Delaunay method (VDM) for DEEP2 data in this field.
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