A.M. Kagan introduced a class of distributions $\mathcal{D}_{m, k}$ in $\mathbb{R}^m$ and proved that if the joint distribution of $m$ linear forms of $n$ independent random variables belongs to the class $\mathcal{D}_{m, m-1}$, then the random variables are Gaussian. A.M. Kagan's theorem implies, in particular, the well-known Skitovich--Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of $n$ independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan's theorem is valid.