1997 Putnam A4

Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$

a function such that

$$\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)$$

whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element

$a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism (i.e.

$\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in G$).

设 $G$ 是一个具有身份 $e$ 和 $\phi:G\rightarrow G$ 的群

一个函数使得

$$\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)$$

每当 $g_1g_2g_3=e=h_1h_2h_3$ 时。证明存在一个元素

$a\in G$ 使得 $\psi(x)=a\phi(x)$ 是同态(即

$\psi(xy)=\psi(x)\psi(y)$ 对于所有 $x,y\in G$)。