Day 7

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Section Day 7

This is an outline of the topics we covered in the seventh 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 6/9

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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Compute products and orders of elements in an external direct product of groups
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Classify the structure of groups of order \(2p\text{,}\) where \(p\) is a prime
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State the following mathematical results: Criterion for \(G\times H\) to be Cyclic
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Construct isomorphisms between the groups \(\Z_n\) or \(U(n)\) and external direct products of cyclic groups of smaller orders.
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Algebraist of the Day.

Tadashi Nakayama, 1912-1964

Japanese mathematician; worked at Osaka and Nagoya
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Known for work in representation theory and commutative algebra, especially Nakayama’s Lemma
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Wrote 6 books and 122 papers in his career despite prolonged health problems
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Reminders/Announcements.

Check-In Survey due Thursday 9:30 am - let me know how class is going so far
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Exam 1 Grades this afternoon or tomorrow morning; Exam 1 Reattempt on June 16.
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Theorem 77. Order of \(HK\).

Let \(H, K\) be finite subgroups of a group \(G\text{.}\) Define \(HK = \{hk : h\in H, k\in K\}\text{.}\) Then

\begin{equation*} |HK|=\frac{|H||K|}{|H\cap K|}\text{.} \end{equation*}

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Theorem 78. Groups of Order 2p are Cyclic or Dihedral.

Let \(G\) be a group of order \(2p\text{,}\) where \(p\geq 3\) is prime. Then \(G \isom \Z_{2p}\) or \(G \isom D_{2p}\text{.}\)

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Proof.

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Theorem 79. Criterion for \(G\times H\) Cyclic.

Let \(G, H\) be finite cyclic groups. Then \(G\times H\) is cyclic if and only if \(\gcd(|G|, |H|)=1\text{.}\)

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Proof.

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Corollary 80. Criterion for Cyclic Product of n Groups.

Let \(G_1,\dots,G_n\) be finite cyclic groups. Then \(G_1\times \cdots \times G_n\) is cyclic if and only if \(\gcd(|G_i|, |G_j|)=1\) for all \(i\neq j\text{.}\)

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Corollary 81. Decomposing \(Z_m\) as a Product of Cyclic Groups.

Let \(m=n_1\cdot n_2 \cdot \dots \cdot n_k\text{.}\) Then

\begin{equation*} Z_m \isom \Z_{n_1} \times \Z_{n_2} \times \cdots \times \Z_{n_k} \Longleftrightarrow \gcd(n_i, n_j)=1 \text{ for all } i\neq j\text{.} \end{equation*}

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