Day 10

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Section Day 10

This is an outline of the topics we covered in the tenth 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 Thursday 6/18

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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State and apply the Diamond, Fraction, and Correspondence Theorems
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State the following mathematical results: Normal Subgroups are Kernels, Fundamental Theorem of Finite Abelian Groups
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Classify all Abelian groups of a given order.
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Algebraist of the Day.

Arthur Cayley, 1821-1895

English mathematician; worked as a lawyer for 14 years and published 200 papers during that time
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Developed the idea of an abstract group and group algebra; Cayley tables and Cayley graphs are named after him
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Involved in efforts to allow women to attend Cambridge
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Reminders/Announcements.

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Theorem 104. Correspondence Theorem - Fourth Isomorphism Theorem.

If \(N\) is a normal subgroup of \(G\text{,}\) then there is a one-to-one correspondence between the subgroups of \(G/N\) and the subgroups of \(G\) that contain \(N\text{.}\) That is, every subgroup of \(G/N\) is of the form \(H/N\) for some subgroup \(H\) of \(G\) containing \(N\text{.}\) Further:

Normal subgroups are preserved: \(H/N \normaleq G/N\) if and only if \(H \normaleq G\)
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Index is preserved: \([G/N : H/N] = [G:H]\)
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Conjugacy is preserved: \(H/N\) is conjugate to \(K/N\) in \(G/N\) if and only if \(H\) is conjugate to \(K\) in \(G\)
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Largest subgroup below and smallest subgroup above are preserved: \(H/N \cap K/N= (H\cap K)/N\) and \(\langle H/N, K/N\rangle = \langle H, K\rangle / N\)
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Theorem 105. Diamond Theorem - Second Isomorphism Theorem.

If \(K\) is a subgroup of \(G\) and \(N\) is a normal subgroup of \(G\text{,}\) then

\(\displaystyle KN \leq G\)
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\(\displaystyle N \normaleq KN\)
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\(\displaystyle (K\cap N) \normaleq K\)
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and the quotients \(K/(K\cap N)\) and \(KN/N\) are isomorphic.
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Theorem 106. Fraction Theorem - Third Isomorphism Theorem.

If \(M\) and \(N\) are normal subgroups of \(G\) with \(N\leq M\text{,}\) then \((G/N)/(M/N)\cong G/M\text{.}\)

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Theorem 107. Fundamental Theorem of Finite Abelian Groups.

Every finite Abelian group is isomorphic to a direct product of cyclic groups of prime power order. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.

\begin{equation*} G \isom \Z_{p_1^{n_1}} \times \Z_{p_2^{n_2}} \times \dots \times \Z_{p_k^{n_k}} \end{equation*}

where the \(p_i\) are not necessarily distinct primes and the \(n_i\) are positive integers.

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Example 108. Abelian Groups of Order 16.

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Example 109. Abelian Groups of Order 72.

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Proof Strategy.

Break the proof into five lemmas, plus Cauchy’s Theorem: if \(p \mid |G|\text{,}\) then \(G\) has an element of order \(p\text{.}\)

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Lemma 110.

Let \(|G|=mn\) with \(\gcd(m,n)=1\) and \(H, K \leq G\) with \(|H|=m\) and \(|K|=n\text{.}\) Then \(G \isom H \times K\text{.}\)

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Lemma 111.

If \(|G|=p_1^{n_1}p_2^{n_2}\dots p_k^{n_k}\) where \(p_i\) are distinct primes and

\begin{equation*} G_{p_i} = \{x \in G \mid x^{p_i^{n_i}}=e\}\text{,} \end{equation*}

then \(G\isom G_{p_1}\times G_{p_2}\times \dots G_{p_k}\) and \(|G_{p_i}|=p_i^{n_i}\text{.}\)

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Lemma 112.

If \(G\) is a \(p\)-group and \(a\) is an element of maximum order in \(G\text{,}\) then \(G \isom \subgroup{a}\times H\) for some \(H \leq G\text{.}\)

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Lemma 113.

An Abelian \(p\)-group is isomorphic to a direct product of cyclic groups of prime power order.

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Lemma 114.

Let \(G\) be an Abelian \(p\)-group. If

\begin{equation*} H_1\times H_2 \dots H_m \isom G \isom K_1 \times K_2 \dots K_n \end{equation*}

where each \(H_i, K_i\) are nontrivial cyclic groups of decreasing order (\(|H_i|\geq |H_{i+1}|\) and \(|K_i|\geq |K_{i+1}|\)), then \(m=n\) and \(|H_i|=|K_i|\) for all \(i\text{.}\)

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