= Solution
A <group homomorphism> preserves multiplication:
$$
\boxed{\Phi(g_1g_2)=\Phi(g_1)\Phi(g_2)\quad\text{for every }g_1,g_2\in G.}
$$
It automatically sends the <identity element> to the identity and sends <inverse elements> to inverses. A <group isomorphism> is a \b[bijective group homomorphism]. For finite groups this requires equal <group orders>, and, once those orders agree, either injectivity or surjectivity of the <group homomorphism> is sufficient. Equal orders alone do not suffice.
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