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This first problem is a repeat from Monday last week. Brennan showed much of this work in class. I would like to start Wednesday's class by reviewing this problem. Each step of the proof below should be quite short.
Problem (Introduction To Internal Direct Products)
Consider the group $G\times H$. Let $A=G\times \{e_H\}$ and $B=\{e_G\}\times H$. Prove the following.
- $A\cap B$ contains only the identity element of $G\times H$.
- The set product $AB$ equals the whole group $G\times H$.
- $A$ and $B$ are normal subgroups of $G\times H$.
- $G\approx A$ and $H\approx B$, which means $G\times H=AB\cong A\times B$.
Did you notice what we did above. We started with a group created by using an external direct product. We then found two subgroup $A$ and $B$ of the group, and showed that the group was equal to the set product $AB$ (a product done internally in the group) and at the same time isomorphic to the external direct product $A\times B$. Given any group $G$, we'd like to know when we can write the group as $G=HK\approx H\oplus K$. The first three conditions from the problem above are the key.
Definition (Internal Direct Product Of Two Subgroups)
Suppose that $G$ is a group. If $H$ and $K$ are normal subgroups of $G$ such that $H\cap K=\{e\}$ and $HK=G$, then we say that $G$ is the internal direct product of $H$ and $K$.
Let's now show that given any two subgroups $H$ and $K$ of a group $G$, that if these subgroups are both normal, their intersection is trivial, and their set product is all of $G$, then their external direct product is isomorphic to $G$.
Problem (Internal Direct Products Are Isomorphic To External Direct Products)
If $G$ is the internal direct product of $H$ and $K$, then prove that the internal direct product $G=HK$ is isomorphic to the external direct product $H\times K$.
Click to see a hint.
We need to build a map from one group to the other, and then prove that the map is an isomorphism (a surjective homomorphism with trivial kernel). The map that takes an element $(h,k)\in H\times K$ and sends it to the element $hk\in HK=G$ should be what we need. It should be clear that the map is surjective (why? Start with something in $G=HK$ and produce an element $(h,k)$ that maps to it.) To prove that this map is a homomorphism, we'll have to show that $hk=kh$. To prove that the map is injective, show that the kernel is trivial. The three properties of being an internal direct product will help.
Given any two finite subgroups $H$ and $K$, there is a simple way to compute the size of their set product. You'll see a similar theorem in combinatorics or probability courses. This is counting principles, and your job is to show that if $x=hk$, then for each $y\in H\cap K$, there is another way to write $x=h_yk_y$, and these are the only ways you can create $x$.
Problem (The Number Of Elements In A Set Product)
Suppose that $G$ is a group and that $H$ and $K$ are finite subgroups of $G$. Prove that $|HK|=|H||K|/|H\cap K|$.
The next problem has you practice working with orders of elements in external direct products, and recognizing when two groups are, or are not, isomorphic. Much of what we'll be doing in the next two weeks has to do with determining when two groups are, or are not, isomorphic.
Problem (Which Groups Of Order 60 Are Isomorphic)
Prove or disprove each of the following. Either build an isomorphism, or show that no such isomorphism exists.
- $\mathbb{Z}_4\oplus Z_{15}\approx \mathbb{Z}_{6}\oplus \mathbb{Z}_{10}$
- $\mathbb{Z}_4\oplus \mathbb{Z}_{15}\approx \mathbb{Z}_{20}\oplus \mathbb{Z}_{3}$
- $D_{20}\oplus \mathbb{Z}_{3}\approx D_{60}$ (Recall that $D_{2n}$ is the automorphisms of a regular $n$-gon.)
- $D_{20}\oplus \mathbb{Z}_{3}\approx \mathbb{Z}_{12}\oplus \mathbb{Z}_5$
Click to see a hint.
Consider orders of elements. What's the largest possible order in each group? How many elements of order 2 does each group have? If you find a mismatch between groups, they cannot be isomorphic.
Problem 101 (Homomorphisms Preserve Normal Subgroups)
Suppose that $f:G\to H$ is a homomorphism. Use The Normal Subgroup Test to prove the following:
- If $N$ is normal in $G$, then $f(N)$ is normal in $f(G)$.
- If $B$ is normal in $H$, then $f^{-1}(B)$ is normal in $G$.
At this point, we've proved quite a large collection of facts about homomorphisms and what they preserve. I strongly suggest that you look at pages 202-204 of your text to see a list of these properties. Then read the remarks on page 204.
The following two facts follow immediately from using the properties of homomorphisms. The first fact shows that any time you have a subgroup that contains half the elements of the group, that subgroup must be a normal subgroup.
Problem 102 (Subgroups Of Index 2 Are Normal)
Suppose that $H$ is a subgroup of $G$ with index $|G:H|=2$. Recall that the index of $H$ in $G$ is the number of distinct cosets of $H$ in $G$.
- Build a surjective homomorphism from $G$ to $\mathbb{Z}_2$.
- Show that $H$ is normal in $G$.
This next fact shows that any time you have subgroup in a factor group $G/N$, it corresponds to a subgroup $H$ in $G$ that contains $N$. So if we know that $G/N$ has a subgroup of
Problem 103 (Subgroups Of A Quotient Group Correspond To Subgroups Of The Original Group)
Suppose that $N$ is a normal subgroup of $G$ and that $B$ is a subgroup of $G/N$. Prove the following:
- There exists a subgroup $H\leq G$ such that $H/N=B$.
- If we know that $n=|N|$ and $m=|B|$ (so $n$ is the order of $N$ in $G$, and $m$ is the order of $B$ in $G/N$), then $G$ must have a subgroup $H$ of order $nm$.
As a suggestion, consider the homomorphism $f:G\to G/N$ given by $f(g)=Ng$ and use some properties of homomorphisms.
For more problems, see AllProblems