According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group.

中文翻译：

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关于包含2阶元素的局部紧Abelian群的刻画定理

根据著名的海德定理，实线上的高斯分布的特征是一种独立于随机变量的线性形式的条件分布的对称性。我们研究了这个定理的类似物对某些局部紧Abel群X含有的顺序2的元件证明了如果X含有2阶的元素，这导致一个事实，即一个宽类非高斯分布对X的特征在于由一种独立的随机变量的线性形式的条件分布的对称性给出。而线性形式的系数是一组的拓扑自同构。