【Basic Abstract Algebra】Exercises for Section 3.3 — Homomorphism of groups

1. Find out all possible homomorphism from \(\mathbb Z_7\to\mathbb Z_{12}\).
   Solution: Let \(\varphi\) be such a homomorphism. Since \(\mathbb Z_7\) is a cyclic group, so \(\varphi\) is specified by \(\varphi(\bar1)\). Since \(o(\bar 1)=7\), we have \(o(\varphi(\bar 1))\mid 7\). And \(o(\varphi(\bar1))\mid 12\) by Lagrange's Theorem. Thus, \(o(\varphi(\bar 1))\mid\gcd(7,12)=1\), i.e., \(o(\varphi(\bar 1))=1\), \(\varphi(\bar1)=\bar0\). Therefore, \(\varphi(\bar x)=\bar0\).

2. Let \(A\) be \(m\times n\) matrix. Show that map

   \[\begin{aligned} \varphi:~\mathbb R^n&\to\mathbb R^m\\ a&\mapsto Aa \end{aligned} \]

   is a homomorphism.
   Proof: Clearly, the map is well defined since for any \(a\in\mathbb R^n\), \(\exists! Aa\in\mathbb R^m\), s.t. \(\varphi(a)=Aa\). Note that

   \[\varphi(a+b)=A(a+b)=Aa+Ab=\varphi(a)+\varphi(b). \]

   Thus, \(\varphi\) is a homomorphism.