We study those semigroups S which are E-unitary (left regular) bands of groups. The main result of the paper says that any such S is isomorphic to the semigroup (under the induced multiplication of the direct product \(E\times G\) ) \((E,G,\varphi )=\{(e,a)\in E\times G:a\in e\varphi \},\) where E is the set of idempotents of S, G is the maximum group homomorphic image of S and \(\varphi \) is a function from E into the lattice of all subgroups of G that meets some natural additional condition. Moreover, we give a description of certain fundamental congruences on \((E,G,\varphi )\) . Finally, we characterize the structure of orthodox bands of groups.
