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Inquiry Based Prep Problems
Pick up where you left off in class. I think you got partway through 5. The big idea is the Sylow theorem. The problems that come after it are other applications related to the ideas that built up to the Sylow theorem. If you are not sure where to go with the Sylow theorem, then work on the other problems first. The Sylow theorem is the longest one. Some of the problems below are extremely short.
Problem (The First Sylow Theorem Proof)
Prove the First Sylow Theorem.
Theorem (The First Sylow Theorem)
Suppose $G$ is a finite group and $p^k$ divides $|G|$ for some prime $p$ and integer $k$. Then $G$ contains a subgroup of order $p^k$.
Click to see a hint.
Use induction on on the order of the group. If $|G|=1$, then what do you know? If $|G|=2$, then what do you know. Assume for some $n\in \mathbb{N}$ that the first Sylow theorem is true for all groups of order $k<n$. Then show that the theorem is true if $|G|=n$.
At some point in your work, you'll need to use two problems that we have already shown to be true.
- Cauchy's Theorem for Abelian Groups - If $p$ divides $|G|$, then $G$ has an element of order $p$. This applies to the center of every group, so if $G$ is not Abelian you can still apply the theorem to the center $Z(G)$ of $G$.
- If $G/N$ has a subgroup $B$, then $G$ has a subgroup $H$ with $H/N=B$. If $G$ is finite, then we know $|H|/|N|=|B|$.
What do you do? If $G$ has order $p^k$, then you're done. If $G$ has a proper subgroup of order $p^k$, then you're done. So suppose it does not have a proper subgroup of order $p^k$. What does this mean about $|C(a)|$? In particular, what does this mean about $|G:C(a)|$ which equals $|\text{cl}|$? Can you prove that $p$ must divide this index for each $a\notin Z(G)$? You may have to use the induction hypothesis your work here. If you then use the class equation, you should be able to show that the center of $G$ contains an element $x$ of order $p$. Once you do that, you can create the factor group $G/\left<x\right>$, which has order less than $G$. Here's where you get to use the induction hypothesis again. Problem 7 from Wednesday last week should help you to get to a subgroup in $G$ from knowing things about subgroups in $G/\left<x\right>$. I believe Nick presented this in class on Monday (at the very end).
Problem (If The Factor Group Of G By The Center Is Cyclic Then G Is Abelian)
Suppose that $G$ is a group and $Z(G)$ is the center of $G$. Prove that if $G/Z(G)$ is cyclic, then $G$ is Abelian.
Problem (Groups Of Order $pq$)
Suppose that $G$ has order $pq$ for primes $p$ and $q$. If $G$ is not Abelian, then what does the previous problem tell you about the center of $G$?
Problem (Groups Of Order $p^2$ Are Abelian)
Prove that every group of order $p^2$ is Abelian.
Definition (Automorphisms And Inner Automorphisms)
Let $G$ be a group.
- An automorphism of $G$ is an isomorphism from $G$ to $G$.
- We write $\text{Aut}(G)$ to represent the set of all automorphisms of $G$.
- The function $\phi_g:G\to G$ defined by $\phi_g(x)=gxg^{-1}$ is called an inner automorphism of $G$.
- We write $\text{Inn}(G)$ to represent the set of all inner automorphisms of $G$.
Problem ($\text{Inn}(G)$ is a normal subgroup of $\text{Aut}(G)$)
Let $G$ be a group. Prove the following:
- $\text{Aut}(G)$ is a group,
- $\text{Inn}(G)$ is a subgroup of $\text{Aut}(G)$, and
- $\text{Inn}(G)$ is a normal subgroup of $\text{Aut}(G)$.
Problem ($G/Z(G)$ is isomorphic to $\text{Inn}(G)$)
Suppose that $G$ is a group. Let $f:G \to \text{Inn}(G)$ be defined by $f(x)=\phi_x$.
- Show that $f$ is a homomorphism with kernel $Z(G)$.
- Then prove that $G/Z(G)$ is isomorphic to $\text{Inn}(G)$.
- Compute $\text{Inn}(G)$ for any Abelian group $G$ and then for $G=D_8$.
- Why is $D_{10}$ isomorphic to $\text{Inn}(D_{10})$?
For more problems, see AllProblems