MATH 415 Lecture 12

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End Exam 1 content

Exam on Tuesday.

Group Factorization

Normal Subgroups

Normal Subgroups

Any of the following conditions are equivalent:

1. $aH=Ha$ for all $a\in G$.
2. $gHg^{-1}=H$ for all $g\in G$
3. $ghg^{-1}\in H$ for all $g\in G$ and $h\in H$.

$h\to ghg^{-1}$ is called the conjugation of $h$.

Let's call this function $i_{g}:H\to H$. It happens that this function also works over $G$ (since $G\trianglelefteq G$, that is, $i_{g}:G\to G$, where $i_{g}(a)=g\,a\,g^{-1}$ for all $a\in G$.

Automorphisms

Automorphisms of $G$ are of the form: $\phi :G\to G$, where $\phi$ is an isomorphism. The set of Automorphisms is a group over composition

Theorem. From our example above, $i_{g}$ is a member of the automorphisms of $G$:

1. It is injective: $i_{g}(a)=i_{g}(b)$ implies $g\,a\,g^{-1}=g\,b\,g^{-1}$, in particular $a=b$.
2. It is surjective: $i_{g}(g^{-1}\,a\,g)=a$. Thus if $i_{g}(b)=a$ if $b=g^{-1}\,a\,g$
3. $i_{g}$ is a homomorphism: $i_{g}(a\,b)=g\,a\,b\,g^{-1}=g\,a\,e\,b\,g^{-1}=(g\,a\,g^{-1})\,(g\,b\,g^{-1}$

Hence $i_{g}$ is an isomorphism.

Inner and Outer Automorphisms

If we have $f,g\in G$, then $i_{f}\circ i_{g}=f(g\,a\,g^{-1})f^{-1}=(f\,g)\,a\,(f\,g)^{-1}=id_{f\,g}$.

The group $\mathrm {Inn} \,G=\left\{i_{g}~\mid ~g\in G\right\}$ is a subgroup of $\mathrm {Aut} (G)$ with identity element $i_{e}=id$. We call this group the inner automorphisms of $G$

Inn(G) is a normal subgroup of Aut(G), so Aut(G)/Inn(G) exists, and we wil call this group the outer automorphisms of G

Going batk to our above criteria, we add the following:

1. $aH=Ha$ for all $a\in G$.
2. $gHg^{-1}=H$ for all $g\in G$ if and only if $i_{g}[H]=G$
3. $ghg^{-1}\in H$ for all $g\in G$ and $h\in H$.
4. $H$ is invariant with respect to inner automorphisms

For a group $G$, we can construct $G$ as the disjoint union of all left cosets $aH$, where $a\in T$ for transversal $N$.

It was mentioned last time that

• $G/\left\{e\right\}\simeq G$
• $G/G\simeq \left\{e\right\}$
• If $G\times L\times K$ and $H\trianglelefteq G$, $L\trianglelefteq L_{1}$, and $K\trianglelefteq K_{1}$, then $G/H\simeq L/L_{1}\times K/K_{1}$

If $H<G$, then the index $(G:H)={\frac {\left|G\right|}{\left|H\right|}}$ is equivalent to the number of cosets.

• If $G$ is finite, then the index must be finite.
• If $G$ is infinite, then the index may or may not be finite.

In the case where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G : H) = 2} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \lefttriangle G} is normal. In particular, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle eH} and $aH$, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ne e} .

This does not hold necessarily for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G : H) = 3} , etc.

---

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = S_n} be the symmetric group, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = A_n} be the alternating group. By definition, $\sigma \in A_{n}$ if and only if $\sigma$ can be expressed as a product of an even number of transpositions.

Therefore $S_{n}\setminus A_{n}$ consists of all odd permutations, and $(S_{n}:A_{n})=2$. Thus $A_{n}\triangleleft S_{n}$ because we can express $S_{n}$ as the disjoint union of $A_{n}$ and $\tau \,A_{n}$ ($S_{n}=A_{n}\sqcup \tau \,A_{n}$) for any $\tau \not \in A_{n}$.

Equivalently, $\left|S_{n}/A_{n}\right|=2$, so $S_{n}/A_{n}$ is isomorphic to $\mathbb {Z} _{2}$.

---

Let $G=\mathbb {Z} _{4}\times \mathbb {Z} _{6}$ is abelian, and let $H=\left\langle (0,1)\right\rangle$ be the cyclic subgroup generated by $(0,1)$. Note that $H\triangleleft G$.

$|G|=26$ because $\mathbb {Z} _{4}$ and $\mathbb {Z} _{6}$ have 4 and 6 elements respectively.

$|H|=6$ because $H=\left\{(0,0),(0,1),\ldots ,(0,5)\right\}$

Therefore $\left|G/H\right|={\frac {|G|}{|H|}}={\frac {24}{6}}=4$.

This example calls to mind that any group of the form $H=\left\{0\right\}\times \mathbb {Z} _{6}$, so $G/H=\mathbb {Z} _{4}/\left\{0\right\}\times \mathbb {Z} _{6}/\mathbb {Z} _{6}\simeq \mathbb {Z} _{4}\times \left\{0\right\}\simeq \mathbb {Z} _{4}$.

What about $H=\left\langle (2,3)\right\rangle$? Then $H=\left\{(2,3),(0,0)\right\}$, so $H\simeq \mathbb {Z} _{2}$. Then $\left|G/H\right|={\frac {24}{2}}=12$.

We know that this group is abelian by the #Theorem, so it is isomorphic to one of the following groups (only choices for abelian groups of order 12):

1. $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{3}$
2. $\mathbb {Z} _{4}\times \mathbb {Z} _{3}$

The main difference between the first group and the second is that the first has no element of order 4. In fact, its only maximal orders are 2, 3, and 2×3=6.

Therefore, our group $G/H$ must be isomorphic to $\mathbb {Z} _{4}\times \mathbb {Z} _{3}$.

Theorem

Theorem. A factor of an abelian group is also abelian: Given an abelian group $G$ and $H<G$, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / H} is abelian.

Proof. We can find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (aH) \, (bH) = (bH)(aH)} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H} because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ab)H = (ba)H} , which is true by the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is abelian.

quod erat demonstrandum

Theorem. A factor group of a cyclic group is also cyclic. Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \left\langle a \right\rangle} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \left\langle a^s \right\rangle} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H} is also cyclic.

Proof. $G/H=\left\langle aH\right\rangle =\left\{H,aH,a^{2}H\right\}$. Observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ a^n \right\} = G} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H} can be expressed in cyclic form generated by the coset containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} .

quod erat demonstrandum

Example 5.12

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \mathbb{Z} \times \mathbb{Z}} and $H=\left\langle (1,1)\right\rangle$. What is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H} ?

Semantically, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} represents the set of points with integer coordinates, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} represents the points along the diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = x} .

The cosets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle aH} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} will represent the diagonal lines that are parallel to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} with integer points.

The transversal can be formed from any line perpendicular to the cosets. Therefore the index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G:H)} is infinite, and its quotient is infinite. In particular, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/H \simeq \mathbb{Z}} .

Exam Review

Bring your own paper.

Calculator is not needed, but if wanted, only 4-function

Start with name.

• Complex Numbers, roots of unity, etc
• Binary Structures: set with binary operation
  • Isomorphism of binary structure
  • Identity element of binary structure
• Groups and Subgroups (axioms and definitions)
  • Def of Subgroup: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_G \in H} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\,b \in H} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b\in H} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in H} implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in H}
• Cyclic Groups and Subgroups: groups that can be generated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in G} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \left\langle a \right\rangle = \left\{ a^n ~\mid~ n \in \mathbb{Z} \right\}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^n = e}
  • Orders of elements: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{order}(a) = n} (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} does not exist
  • Note that the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} (or the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle a \right\rangle} ) is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{order}(a) = \left| \left\langle a \right\rangle \right|} , not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a|} !
  • In all cases, either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle a \right\rangle < G} is a proper subgroup or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle a \right\rangle = G} generates the group.
  • Cyclic subgroups are isomorphic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} if infinite, and isomorphic to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_n} if finite and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| G \right| = n} .
  • A subgroup of cyclic group is also cyclic.
  • Subgroups of cyclic groups can always be expressed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ 0 \right\}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \mathbb{Z}}
  • By Theorem 6.14, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H < \mathbb{Z}_n}
  • If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a finite cyclic group generated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , and $H<G$ is a cyclic subgroup generated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^s} , then for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \gcd(s,n)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| H \right| = \frac{n}{d}} .
• Permutations, symmetric group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n} from all permutations of $n$ elements, and alternating group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_m} from all even permutations
  • cycles, orbits, etc.
• Direct Products
  • Order of element of direct product is LCM of each element in respective group.
• Fundamental Theorem of Finitely Generated Abelian Groups
• Normal Subgroups (definitions)
  • Homomorphism, Kernel
• Factor Groups
  • Definition of product of cosets