# HW 4, due Wed Feb 8, 2017

• DF Section 2.4 #9 #17 (Zorn’s lemma appears in Appendix 1), #18
• DF Section 3.1 #1, #12.
• Let F be any field.  Show that there is a 1-1 homormophism from $S_n$ to $GL(n,F)$ given by the action of $S_n$ on the indices i in the standard basis $e_1,\ldots,e_n$.  Combine this with Cayley’s theorem (page 120) to show that for every finite group G, there exists n, such that G is isomorphic to a subgroup of $GL(n,F)$.
• Show algebraically that every isometry T of $R^2$ is a glide reflection provided it has the form $\latex T=T_\lambda A$, where $A\in O(2,R)$, $A\not\in SO(2,R)$, $\latex \lambda\ne 0$.
• Solve the following problems without submitting a solution
• DF Section 2.4 #15, #19