Day 5

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Section Day 5

This is an outline of the topics we covered in the fifth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.

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Handout Tuesday 6/2

Objectives: Advanced Learning Outcomes

During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.

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Given a map between groups, determine if it is an isomorphism
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Given two isomorphic groups, find an isomorphism between them
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State the following mathematical results: Cayley’s Theorem, Isomorphisms Preserve Element Properties, Isomorphisms Preserve Group Properties
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Use the definition of isomorphism to prove results about the properties of isomorphisms.
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State and instantiate the definitions of: automorphism, inner automorphism
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See how automorphisms of a group act on a Cayley diagram of that group.
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Algebraist of the Day.

Niels Abel, 1802-1829

Norwegian mathematician: independently from Galois developed group theory and proved there is no general formula for solving quintic equations by radicals
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Laid the foundations of elliptic function theory
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Abelian groups are named after him and the Abel prize is awarded in his honor
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Tragically died at 26 from tuberculosis
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Reminders/Announcements.

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Definition 49. Alternating Group.

The alternating group on \(n\) elements, denoted \(A_n\text{,}\) is the subgroup of the symmetric group \(S_n\) consisting of all even permutations.

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Theorem 53. Cayley’s Theorem.

Every group is isomorphic to a permutation group.

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Theorem 54. Isomorphisms Preserve Element Properties.

Let \(\phi: G\to \overline{G}\) be an isomorphism of groups, \(a,b \in G\text{,}\) and \(k\in \Z\text{.}\) Then

\(\displaystyle \phi(e_G)=e_{\overline{G}}\)
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\(\displaystyle \phi(a^k)=(\phi(a))^k\)
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\(\displaystyle ab=ba \Longleftrightarrow \phi(a)\phi(b)=\phi(b)\phi(a)\)
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\(\displaystyle G=\subgroup{a} \Longleftrightarrow \overline{G}=\subgroup{\phi(a)}\)
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\(\displaystyle |a|=|\phi(a)|\)
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\(x^k=b\) in \(G\) has the same number of solutions as \(y^k=\phi(b)\) in \(\overline{G}\text{.}\)
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If \(|G|=n\text{,}\) then \(G\) and \(\overline{G}\) have the same number of elements of each order.
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Theorem 55. Isomorphisms Preserve Group Properties.

Let \(\phi: G\to \overline{G}\) be an isomorphism of groups. Then

\(\phi^{-1}\) is an isomorphism from \(\overline{G}\) to \(G\text{.}\)
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\(G\) is Abelian if and only if \(\overline{G}\) is Abelian.
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\(G\) is cyclic if and only if \(\overline{G}\) is cyclic.
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\(\displaystyle H \leq G \Longleftrightarrow \phi(H):=\{\phi(h)\mid h \in H\} \leq \overline{G}\)
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\(\displaystyle \overline{H} \leq \overline{G} \Longleftrightarrow \phi^{-1}(\overline{H}):=\{g \in G \mid \phi(g) \in \overline{H}\} \leq G\)
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\(\displaystyle \phi(Z(G))=Z(\overline{G})\)
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Partial Proof of Isomorphisms Preserve Group Properties.

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Example 56. Automorphisms and Cayley Diagrams.

Let’s think about how automorphisms of a group act on the Cayley diagram of that group, using \(\Z_5\) as an example, with the map \(\varphi(x)=2x\pmod 5\text{.}\)

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