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Math 113 Homework 5. Due 10/6
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\noindent \textbf{Reading corresponding to lectures:} \\
- Tuesday 9/29: DF 3.1, 3.2 \\
- Thursday 10/1 DF 3.2, 3.3. You are not responsible for knowing the material in 3.4, but we will talk about it. We won't cover 3.5 -- you did the most important part on problem set 4!
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\noindent \textbf{Problems to hand in:}
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\item (The relation ``is a normal subgroup of" is not transitive). Let $G = D_8$. Show that $\langle s, r^2 \rangle$ is a normal subgroup of $D_8$ and that $\langle s \rangle$ is a normal subgroup of $\langle s, r^2 \rangle$, but $\langle s \rangle$ is \emph{not} a normal subgroup of $D_8$.
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\item DF section 3.1: 14 parts a and b only, 16, 20.
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\item Do the following problems from DF section 3.2: 5, 11, 12. For problem 11, you may assume that $|G:H|$ and $|H:K|$ are finite, but not that $G$ is finite!
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\item Let $H$ and $K$ be subgroups of $G$, with $H \neq K$.
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\item Prove that if $|H|$ and $|K|$ are relatively prime, then $H \cap K = \{e\}$.
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\item Prove that if $|H| = |K| = p$, where $p$ is prime, then $H \cap K = \{e\}$.
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\item Suppose $G$ is a group with $|G| = 35$. Show that $G$ has at most 8 subgroups of order 5, and at most 5 subgroups of order 7. (hint: use part b)
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\item Using the above, show that if $|G| = 35$, then $G$ contains an element of order 5. (hint: if $G$ is cyclic, then $G = Z_{35}$ which has an element of order 5. If $G$ is not cyclic, what are the possible orders of elements? Why can't every element have order 7?)
Now show using the same kind of argument that $G$ has an element of order 7.
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\textit{Remark: This is not just a special fact about groups of order 35 -- Cauchy's theorem (Theorem 11 in DF 3.2) says that, for each prime dividing the order of a group $G$, there is an element in $G$ of that order. }
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\item (Orbits and cosets)
Let $G$ act on a set $A$. For each $a \in A$, let $\stab(a) = \{g \in G \mid g\cdot a = a\}$. We showed earlier that $\stab(a)$ was a subgroup.
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Let $a \in A$ and $x \in G$. Show that the set of left cosets of $\stab(a)$ are in one-to-one correspondence with elements of the orbit of $a$. (show that $\psi: x \stab(a) \mapsto x \cdot a$ is well defined and a bijection).
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\item (not to hand in:) use this to compare the two proofs we gave of Lagrange's theorem
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\noindent The next two questions use material that will be discussed on Thursday 10/1:
\item Use Fermat's little theorem to prove the following: Let $n_1, n_2, ... n_{30}$ be integers, and assume that at least one of them is not divisible by 31. \\Show that
$(n_1)^{30} + (n_2)^{30}+ ... + (n_{30})^{30}$ is also not divisible by 31.
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\item Let $G = GL_2(\C)$. You may take it as a fact that $Z(G) = \{ \lambda I \mid \lambda \in \C-\{0\} \, \}$, the multiples of the identity matrix.
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\item Let $A = SL_2(\C)$ be the set of matrices with determinant 1. Let $\phi$ be the determinant homomorphism, $\phi: GL_2(\C) \to \C^{\times}$. Use the first isomorphism theorem to describe $G/A$.
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\item Now let $B = Z(G)$. Show that $G=AB$.
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What is $A \cap B$?
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\item What can you conclude using the second isomorphism theorem (what group is isomorphic to $G/B$?)
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Remark: This group is called the \emph{projective general linear group}, written $PGL_2(\C)$ or $PSL_2(\C)$ (the fact that it has two names comes from the isomorphism above!)
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