中国剩余定理

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Wikipedia
整数 $m_1,m_2$ 互质，则对任意的整数$a_1,a_2$，同余方程组$\left\{\begin{array}{l}x \equiv a_{1}\pmod {m_1}\\ x \equiv a_{2}\pmod {m_2}\end{array}\right.$有解.
一般地，见rings The Chinese Remainder Theorem
Remark 2.19.
群$G$的正规子群$H_1,H_2$，满足$H_1H_2=G$，则
$$f:G\to(G/H_1)\times(G/H_2);\qquad x\mapsto(xH_1,xH_2)$$为满射，即使$\operatorname{codom}f$不一定是群.
如何证明呢

hbghlyj

对$(G/H_1)\times(G/H_2)$的任何元素$(a_1H_1,a_2H_2)$, 设$x=a_1a_2^{-1}a_1^{-1}a_2$, 则
\begin{align*}xH_1&=a_1(a_2^{-1}a_1^{-1}a_2)H_1=a_1H_1\\xH_2&=H_2x=H_2(a_1a_2^{-1}a_1^{-1})a_2=H_2a_2=a_2H_2\end{align*}故$f(x)=(a_1H_1,a_2H_2)$.