## 数学代写|抽象代数作业代写abstract algebra代考|Cayley’s Theorem and its Applications

In earlier sections, we mentioned the difficulty of describing all groups and their elements in a consistent manner. Cayley’s Theorem offers one approach.

Before giving the proof of this theorem and illustrating it with examples, we mention the importance for computational methods. Because of Cayley’s Theorem implementing operations and methods for permutation groups (subgroups of some $S_n$ ) ultimately encompasses all finite groups. For example, in SAGE, the command G=DihedralGroup (5) defines the group $G$ as the subgroup of $S_5$ expressed as
$$\langle(12345),(25)(34)\rangle .$$
In particular, every finite group is isomorphic to one constructed as described in Subsection 1.7.3.

Both Maple (GRoupTHEORY package) and SaGE offer many commands that construct spécific groups. For example, both have commands for the nth dicyclic group, for the Baby Monster Group, for a projective symplectic group over a finite field, and many more.

Proof (of Theorem 1.9.27). Write the elements of $G$ as $G=\left{g_1, g_2, \ldots, g_n\right}$. Define the function $\psi: G \rightarrow S_n$ by $\psi(g)=\sigma$ where
$$g g_i=g_{\sigma(i)} \quad \text { for all } i \in{1,2, \ldots, n}$$

## 数学代写|抽象代数作业代写abstract algebra代考|Generators and Relations

In Section 1.1.2, where we introduced the abstract notation for the dihedral group $D_n$, we saw that all dihedral symmetries can be obtained as various compositions of $r$ (the rotation by angle $2 \pi / n$ ) and $s$ (the reflection through the $x$-axis). We also know that $r^n=1$ because $r$ has order $n$, and that $s^2=1$ because $s$ has order 2 . However, we also proved that $r$ and $s$ satisfy $s r=r^{-1} s$.
Since every element in $D_n$ can be created from operations on $r$ and $s$, we say that $r$ and $s$ generate $D_n$ and that $r^n=1, s^2=1$, and $s r=r s^{-1}$ are relations on $r$ and $s$. In group theory, we write
$$D_n=\left\langle r, s \mid r^n=s^2=1, s r=r^{-1} s\right\rangle$$
to express that every element in $D_n$ can be obtained by a finite number of repcated opcrations betwoen $r$ and $s$ and that cvery algebraic relation betwecn $r$ and $a$ can be deduced from the relations $r^n=1, a^2=1$, and $a r=r^{-1} a$. The expression (1.9) is the standard presentation of $D_n$.

The relation $s r=r^{-1} s$ shows that in any term $s^k r^l$ it is possible to “move” the $s$ to the left of the term by appropriately changing the power on $r$. Hence, in any word involving the generators $r$ and $s$, it is possible, by appropriate changes on powers, to move all the powers of $s$ to the right and all the powers of $r$ to the left. Hence, every expression in the $r$ and $s$ can be rewritten as $r^l s^k$ with $0 \leq k \leq 1$ and $0 \leq l \leq n-1$. Though we already knew $\left|D_n\right|=2 n$ from geometry, this reasoning shows that there are at most $2 n$ terms in the group given by this presentation.

# 抽象代数代考

## 数学代写|抽象代数作业代写抽象代数代考|Cayley定理及其应用

$$\langle(12345),(25)(34)\rangle .$$

Maple (GRoupTHEORY包)和SaGE都提供了许多构造spécific组的命令。例如，它们都有针对第n个双环群的命令，针对小怪物群的命令，针对有限域上的投影辛群的命令，以及更多的命令

$$g g_i=g_{\sigma(i)} \quad \text { for all } i \in{1,2, \ldots, n}$$

## 数学代写|抽象代数作业代写abstract algebra代考|Generators and Relations

$$D_n=\left\langle r, s \mid r^n=s^2=1, s r=r^{-1} s\right\rangle$$

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