08-401/Homework Assignment 9

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Reading

Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! You may also go over Chapter 32 of Gallian's book once again; we are not done with that chapter yet.

Doing

Solve and submit the following three problems.

Problem 1. Show that the dihedral groups $D_{n}$ are solvable ($D_{n}$ is the group of symmetries of a perfect $n$-gon, including rotations and reflections).

Problem 2. The "$ax+b$ group" is the group of all functions $f:{\mathbb {R} }\to {\mathbb {R} }$ of the form $f(x)=ax+b$, with $a,b\in {\mathbb {R} }$ and with $a\neq 0$, under the operation of composition of functions. Show that the $ax+b$ group is solvable.

Problem 3. Show that any subgroup of a solvable group is solvable.

Due Date

This assignment is due in class on Wednesday April 2, 2008.