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content/Math4302/Math4302_L5.md

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+# Math4302 Modern Algebra (Lecture 5)
+
+## Groups
+
+### Subgroups
+
+A subset $H\subseteq G$ is a subgroup of $G$ if
+
+- $e\in H$
+- $\forall a,b\in H, a b\in H$
+- $a\in H\implies a^{-1}\in H$
+
+_$H$ with $*$ is a group_
+
+We denote as $H\leq G$.
+
+<details>
+<summary>Example</summary>
+
+For an arbitrary group $(G,*)$,
+
+$(\{e\},*)$ and $(G,*)$ are always subgroups.
+
+---
+
+$(\mathbb{Z},+)$ is a subgroup of $(\mathbb{R},+)$.
+
+---
+
+Non-example:
+
+$(\mathbb{Z}_+,+)$ is not a subgroup of $(\mathbb{Z},+)$.
+
+---
+
+Subgroup of $\mathbb{Z}_4$:
+
+$(\{0,1,2,3\},+)$ (if $1\in H$, $3\in H$)
+
+$(\{0,2\},+)$
+
+$(\{0\},+)$
+
+---
+
+Subgroup of $\mathbb{Z}_5$:
+
+$(\{0,1,2,3,4\},+)$
+
+$(\{0\},+)$
+
+_Cyclic group with prime order has only two subgroups_
+
+---
+
+Let $D_n$ denote the group of symmetries of a regular $n$-gon. (keep adjacent points pairs).
+
+$$
+D_n=\{\sigma\in S_n\mid i,j\text{ are adjacent } \iff \sigma(i),\sigma(j)\text{ are adjacent }\}
+$$
+
+$$
+\begin{pmatrix}
+1&2&3&4\\
+2&3&1&4
+\end{pmatrix}\notin D_4
+$$
+
+$D_4$ has order $8$ and $S_4$ has order $24$.
+
+$|D_n|=2n$. ($n$ option to rotation, $n$ option to reflection. For $\sigma(1)$ we have $n$ option, $\sigma(2)$ has 2 option where the remaining only has 1 option.)
+
+Since $1-4$ is not adjacent in such permutation.
+
+$D_n\leq S_n$ ($S_n$ is the symmetric group of $n$ elements).
+
+</details>
+
+#### Lemma of subgroups
+
+If $H\subseteq G$ is a non-empty subset of a group $G$.
+
+then ($H$ is a subgroup of $G$) if and only if ($a,b\in H\implies ab^-1\in H$).
+
+<details>
+<summary>Proof</summary>
+
+If $H$ is subgroup, then $e\in H$, so $H$ is non-empty and if $a,b\in H$, then $b^{-1}\in H$, so $ab^{-1}\in H$.
+
+---
+
+If $H$ has the given property, then $H$ is non-empty and if $a,b\in H$, then $ab^-1\in H$, so
+
+- There is some $a,a\in H$, $aa^{-1}\in H$, so $e\in H$.
+- If $b\in H$, then $e\in H$, so $eb^{-1}\in H$, so $b^{-1}\in H$.
+- If $b,c\in H$, then $c^{-1}$, so $bc^{-1}^{-1}\in H$, so $bc\in H$.
+
+</details>
+
+#### Cyclic group
+
+$G$ is cyclic if $G$ is a subgroup generated by $a\in G$. (may be infinite)
+
+$\mathbb{Z}_n\leq D_n\leq S_n$.
+
+Cyclic group is always abelian.