  We consider algebras with basis numerated by elements of a group $G.$ We fix
a function $f$ from $G\times G$ to a ground field and give a multiplication of
the algebra which depends on $f$. We study the basic properties of such
algebras. In particular, we find a condition on $f$ under which the
corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup
$\hat G$ of $G$ we define a $\hat G$-periodic algebra, which corresponds to a
$\hat G$-periodic function $f,$ we establish a criterion for the right
nilpotency of a $\hat G$-periodic algebra. In addition, for $G=\mathbb Z$ we
describe all $2\mathbb Z$- and $3\mathbb Z$-periodic algebras. Some properties
of $n\mathbb Z$-periodic algebras are obtained.
