189-245A: Honours Algebra 1

Assignment 5

Due: Monday, November 24.

1. Show that the symmetric group $S_5$ contains a subgroup of cardinality $20$. Use this to construct a subgroup $H$ of $S_6$ which is isomorphic to $S_5$ but acts transitively on $\{1,2,3,4,5,6\}$, i.e., that acts on this set with a single orbit. Show that $H$ is not conjugate in $S_6$ to the ''obvious" copy of $S_5$ consisting of the permutations on $\{1,2,3,4,5\}$ which fix $6$.

2. Show that the group $\bf{GL}_2(\mathbb Z/2\mathbb Z)$ of $2\times 2$ matrices with entries in the field with two elements is isomorphic to the symmetric group $S_3$ on three elements. (Hint: consider the natural action of $G$ on the set of non-zero column vectors in $(\mathbb Z/2\mathbb Z)^2$.)

3. Let $P=\{0,1,2,\infty\} = \mathbb Z/3\mathbb Z \cup \{\infty\}$ be the so-called ``projective line" over $\mathbb Z/3\mathbb Z$. Show that every element of the group $G=\bf{GL}_2(\mathbb Z/3\mathbb Z)$ induces a permutation on $P$, whereby the matrix $ \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$ sends $j\in P$ to $\frac{aj+b}{cj+d}$, with the natural conventions. What is the kernel of the resulting homomorphism $$ \varphi:G \longrightarrow S_P$$ from $G$ to the group $S_P\simeq S_4$ of permutations on $P$? Is $\varphi$ surjective?

4. Let $S$ be a subset of a group $G$. The centraliser of $S$, denoted $Z(S)$, is the set of $a\in G$ which commute with every $s\in S$, i.e., such that $as=sa$ for all $s\in S$. Show that $Z(S)$ is a subgroup of $G$.

5. Recall that the conjugacy class of $a$ in a group $G$ is the set of all elements of $G$ which are of the form $gag^{-1}$ for some $g\in G$. Show that a normal subgroup of $G$ is a disjoint union of conjugacy classes. List the conjugacy classes in $S_4$ and use this to give a complete list of all the normal subgroups of $S_4$. Same question for $S_5$.

6. Write the group $G=S_4$ as a disjoint union of distinct left $H$-cosets for $G/H$, and as a disjoint union of right cosets $H\backslash G$, when

a) $H$ is the dihedral group $D_8$ of order $8$ viewed as asubgroup of $G$ by $$ H = \{ 1, (1234), (13)(24), (1432), (12)(34), (14)(23), (13), (24) \},$$

b) $H$ is the Klein $4$-group defined by $$ H = \{ 1, (12)(34), (13)(24), (14)(23) \}.$$

7. Give a complete list of the subgroups of $D_8$, say which ones are normal, and for each normal subgroup $H$ describe the associated quotient group $G/H$. Same question for $Q_8$.

8. Show that $S_4$ contains a subgroup of index $3$, and use this to construct a non-trivial homomorphism from $S_4$ to $S_3$. What is the kernel of this homomorphism? Conclude that it is surjective.

9. Let $G$ be a finite group and let $p$ be the smallest prime dividing its cardinality. Show that a subgroup $H$ of $G$ of index $p$ in $G$ must be normal, and describe the quotient group $G/H$.

10. Let $G={\bf GL}_2(\mathbb Z)$, fix an odd prime $p$ and let $H$ be the kernel of the natural homomorphism $$ \varphi: G \rightarrow {\bf GL}_2(\mathbb Z/ p\mathbb Z).$$ Show that the group $H$ is torsion free, i.e., that it has no elements of finite order other than the identity. Use this to get a strong upper bound on the possible elements of finite order in $G$. Can you consruct elements of finite order in $G$ whoe orders you have not ruled out?