We investigate the relation between the projected morphology (b/a) and the velocity dispersion (sigma_v) of groups of galaxies using two recently compiled group catalogs, one based on the 2MASS redshift survey and the other on the SDSS Data Release 5 galaxy catalog. We find that the sigma_v of groups is strongly correlated with the group projected b/a and size, with elongated and larger groups having a lower sigma_v. Such a correlation could be attributed to the dynamical evolution of groups, with groups in the initial stages of formation, having small sigma_v, a large size and an elongated shape that reflects the anisotropic accretion of galaxies along filamentary structures. The same sort of correlations, however, could also be reproduced in prolate-like groups, if the net galaxy motion is preferentially along the group elongation, since then the groups oriented close to the line of sight will appear more spherical, will have a small projected size and large sigma_v, while groups oriented close to the sky-plane will appear larger in projection, more elongated, and will have smaller sigma_v. We perform tests that relate only to the dynamical evolution of groups (eg., calculating the fraction of early type galaxies in groups) and indeed we find a strong positive (negative) correlation between the group sigma_v (projected major axis) with the fraction of early type galaxies. We conclude that (a) the observed dependencies of the group sigma_v on the group projected size and b/a, should be attributed mostly to the dynamical state of groups and (b) groups in the local universe do not constitute a family of objects in dynamical equilibrium, but rather a family of cosmic structures that are presently at various stages of their virialization process.
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