Section 67: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Show that if $G_{1}$ is a subgroup of $G$ , there may be no subgroup $G_{2}$ of $G$ such that $G=G_{1}\bigoplus G_{2}$ . [Hint: Set $G=\mathbb{Z}$ and $G_{1}=2\mathbb{Z}$ .]

We set $G$ and $G_{1}$ as suggested. Suppose $G_{2}$ is a subgroup of $G$ such that $G=G_{1}\bigoplus G_{2}$ . Then, $G_{2}$ contains some odd number $z$ , and, hence, $2z\in G_{2}$ where $2z\neq0$ , which implies that $2z$ has different representations as the sum of elements of $G_{1}$ and $G_{2}$ . Or, for example, $2z+2\in G_{1}$ , and $2z+2=2+2z$ where $2\in G_{1}$ and $2z\in G_{2}$ etc.

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Section 67: Problem 2 Solution

Section 67: Problem 2 Solution