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135 lines
4.1 KiB
Markdown
135 lines
4.1 KiB
Markdown
# Math4303 Modern Algebra (Lecture 12)
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## Groups
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### Direct products
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$\mathbb{Z}_m\times \mathbb{Z}_n$ is cyclic if and only if $m$ and $n$ have greatest common divisor $1$.
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More generally, for $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_k}$, if $n_1,n_2,\cdots,n_k$ are pairwise coprime, then the direct product is cyclic.
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<details>
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<summary>Proof</summary>
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For the forward direction, use $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}=\mathbb{Z}_{n_1n_2}$. if $n_1, n_2$ are coprime.
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For the backward, suppose to the contrary that for example $\gcd(n_1,n_2)=d>1$, then $G=\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times H$, where any element in $H$ has order $\leq |H|$ and any element in $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}$ has order $<\frac{n_1n_2}{d}$, therefore, all the elements in $G$ will have order strictly less than the size $n_1n_2\ldots n_k$ of the group.
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</details>
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#### Corollary for composition of cyclic groups
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If $n=p_1^{m_1}\ldots p_k^{m_k}$, where $p_i$ are distinct primes, then the group
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$$
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G=\mathbb{Z}_n=\mathbb{Z}_{p_1^{m_1}}\times \mathbb{Z}_{p_2^{m_2}}\times \cdots \times \mathbb{Z}_{p_k^{m_k}}
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$$
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is cyclic.
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<details>
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<summary>Example for product of cyclic groups and order of element</summary>
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$$
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\mathbb{Z}_{8}\times\mathbb{Z}_8\times \mathbb{Z}_12
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$$
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the order for $(1,1,1)$ is 24.
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What is the maximum order of an element in this group?
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Guess:
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$8*3=24$
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</details>
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### Structure of finitely generated abelian groups
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#### Theorem for finitely generated abelian groups
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Every finitely generated abelian group $G$ is isomorphic to
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$$
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Z_{p_1}^{n_1}\times Z_{p_2}^{n_2}\times \cdots \times Z_{p_k}^{n_k}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}}
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$$
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<details>
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<summary>Example</summary>
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If $G$ is abelian of size $8$, then $G$ is isomorphic to one of the following:
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- $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$ (non cyclic)
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- $\mathbb{Z}_2\times \mathbb{Z}_4$ (non cyclic)
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- $\mathbb{Z}_2$ (cyclic)
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And any two of them are not isomorphic
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---
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Find all abelian group of order $72$.
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Since $72=2^3*3^2$, There are 3 possibilities for the $2^3$ part, and there are 2 possibilities for the $3^2$ part.
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Note that $\mathbb{Z}_8\times\mathbb{Z}_9$, where $8,9$ are coprime, $\mathbb{Z}_8\times\mathbb{Z}_9=\mathbb{Z}_{72}$, is cyclic.
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There are 6 possibilities in total.
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</details>
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#### Corollary for divisor size of abelian subgroup
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If $g$ is abelian and $|G|=n$, then for every divisor $m$ of $n$, $G$ has a subgroup of order $m$.
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> [!WARNING]
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>
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> This is not true if $G$ is not abelian.
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>
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> Consider $A_4$ (alternating group for $S_4$) does not have a subgroup of order 6.
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<details>
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<summary>Proof for the corollary</summary>
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Write $G=\mathbb{Z}_{p_1}^{n_1}\times \mathbb{Z}_{p_2}^{n_2}\times \cdots \times \mathbb{Z}_{p_k}^{n_k}$ where $p_i$ are distinct primes.
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Therefore $n=p_1^{m_1}\ldots p_k^{m_k}$.
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For any divisor $d$ of $n$, we can write $d=p_1^{m_1}\ldots p_k^{m_k}$, where $m_i\leq n_i$.
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Now for each $p_i$, we choose the subgroup $H_i$ of size $p_i^{m_i}$ in $\mathbb{Z}_{p_i}^{n_i}$. (recall that every cyclic group of size $r$ and any divisor $s$ of $r$, there is a subgroup of order $s$. If the group is generated by $a$, then use $a^{\frac{r}{s}}$ to generate the subgroup.)
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We can construct the subgroup $H=H_1\times H_2\times \cdots \times H_k$ is the subgroup of $G$ of order $d$.
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</details>
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### Cosets
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#### Definition of Cosets
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Let $G$ be a group and $H$ its subgroup.
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Define a relation on $G$ and $a\sim b$ if $a^{-1}b\in H$.
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This is an equivalence relation.
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- Reflexive: $a\sim a$: $a^{-1}a=e\in H$
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- Symmetric: $a\sim b\Rightarrow b\sim a$: $a^{-1}b\in H$, $(a^{-1}b)^{-1}=b^{-1}a\in H$
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- Transitive: $a\sim b$ and $b\sim c\Rightarrow a\sim c$ : $a^{-1}b\in H, b^{-1}c\in H$, therefore their product is also in $H$, $(a^{-1}b)(b^{-1}c)=a^{-1}c\in H$
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So we get a partition of $G$ to equivalence classes.
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Let $a\in G$, the equivalence class containing $a$
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$$
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aH=\{x\in G| a\sim x\}=\{x\in G| a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\}
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$$
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This is called the coset of $a$ in $H$.
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<details>
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<summary>Example</summary>
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Consider $G=S_3$
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</details> |