Exercises

1. Let $G = GL(n,\mathbb{R})$ and $N = SL(n,\mathbb{R})$ for $n\geq 1$.

   (a) Let $A \in G$ and show that the coset $AN = \{X \in GL(n,þ) \ \vert\ \det X = \det A\}$.

   (b) Let $\mathbb{R}^\times$ denote the group of non-zero real numbers with respect to the usual multiplication. Show $G/N\cong \mathbb{R}^\times$.

   (c) Is $G/N\cong G$? Why or why not?

2. Let $G = \langle a\rangle$ be a cyclic group. Let $H$ be any subgroup of $G$.

   (a) Show that $H = \langle a^s\rangle$, where $s$ is an arbitrary positive integer if $\vert G\vert=\infty$, while if $\vert G\vert < \infty$, then $s\vert \vert G\vert$.

   HINT: Go back to §5.2 on cyclic groups.

   (b) Explain why $H \lhd G$. Show that $G/H = \{H, aH, ..., a^{s-1}H\} = \langle aH\rangle$ . In words this says, that a factor group of a cyclic group is cyclic.

3. Using your result from exercise 7 of Section 6.1, write the Cayley table for $A_4/V_4$.

4. Prove that $A_4$ has no subgroup of order $6$.

   HINT: Assume it does. Let $H \leq A_4$ with $\vert H\vert = 6$. Then $H \lhd A_4$ (Why?). So $A_4/H$ makes sense. Moreover, this also implies that $f^2 \in H$ for all $f \in A_4$ (Why?). Now look at the table for $A_4$ and count the number of squares to come to a contradiction.) (Note this problem gives an example to show that the converse of Lagrange's Theorem is false.