  In this paper we study the $\kappa$-word problem for the pseudovariety ${\bf
LG}$ of local groups, where $\kappa$ is the canonical signature consisting of
the multiplication and the pseudoinversion. We solve this problem by
transforming each arbitrary $\kappa$-term $\alpha$ into another one called the
canonical form of $\alpha$ and by showing that different canonical forms have
different interpretations over ${\bf LG}$. The procedure of construction of
these canonical forms consists in applying elementary changes determined by a
certain set $\Sigma$ of $\kappa$-identities. As a consequence, $\Sigma$ is a
basis of $\kappa$-identities for the $\kappa$-variety generated by ${\bf LG}$.
