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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, inparticular, the well-known Skitovich--Darmois theorem, where the Gaussiandistribution on the real line is characterized by independence of two linearforms of $n$ independent random variables. In the note we describe a wide classof locally compact Abelian groups where A.M. Kagan's theorem is valid.

On a generalisation of the Skitovich--Darmois theorem for several linear forms on Abelian groups