Abelian groups

What's abelian and purple? An abelian grape!

Definition 5.6.1   Let $g$ and $h$ be two elements of a group $G$. We say that $g$ commutes with $h$ (or that $g, h$ commute) if $g*h=h*g$. We call a group commutative (or abelian) if every pair of elements $g, h$ belonging to $G$ commute. If $G$ is a group which is not necessarily commutative then we call $G$ noncommutative (or nonabelian).

Example 5.6.2   The integers, with ordinary addition as the group operation, is an abelian group.

Now the reader should understand the punchline to the joke quoted at the beginning! Convention: When dealing with groups in general we often drop the $*$ and denote multiplication simply by juxtaposition (that is, sometimes we write $gh$ in place of $g*h$ and $g^n$ in place of $g*g*...*g$ ($n$ times)). Also, by convention, $g^0=1$, the identity element. However, if the group $G$ is abelian then one often replaces $*$ by $+$, $g+g+...+g$ by $ng$, and then $+$ is not dropped. Some typical examples of finite abelian groups:

• the additive group $G=\mathbb{Z}/n\mathbb{Z}$ (= $C_n$),
• the multiplicative group $G=(\mathbb{Z}/n\mathbb{Z})^\times$,

where $n$ is an integer. We have

$$\vert\mathbb{Z}/n\mathbb{Z}\vert=n, \ \ \ \vert(\mathbb{Z}/n\mathbb{Z})^\times\vert=\phi(n),$$

where $\phi(n)$ is Euler's $\phi$-function.

Definition 5.6.3   If $G$ is a multiplicative group $G$ with only one generator (i.e., there is a $g\in G$ such that $G=\{g^i\ \vert\ i=0,1,2,...\}$) then we say that $G$ is cyclic.

Lemma 5.6.4   If

$$G = \langle g\rangle=\{g^i\ \vert\ i\in \mathbb{Z}\},$$

is a finite cyclic group and $m>1$ is the smallest integer such that $g^m=1$ then $\vert G\vert=m$.

proof: Since $g^{m+k}=g^k$ for any $k\in \mathbb{Z}$, we can list all the elements of $G$ as follows:

$$1, g, g^2, ..., g^{m-1}.$$

There are $m$ elements in this list. $\Box$ We have seen (multiplicative) cyclic groups before in our discussion of the discrete log problem. The abstract form of the discrete log problem is the following: Given a finite cyclic group $G = \langle a\rangle$ with generator $a$ and given $b\in G$, find $x\in \mathbb{Z}$ such that $a^x=b$.

Exercise 5.6.5   Show that any group having exactly 2 elements is abelian.

Exercise 5.6.6   Write down all the elements of $G=(\mathbb{Z}/10\mathbb{Z})^\times$ and compute its multiplication table.

Exercise 5.6.7   Write down all the elements of $G=\mathbb{Z}/4\mathbb{Z}$ and compute it's addition table.

Exercise 5.6.8   Let $SL(2,\mathbb{Z})$ be the subset of all $g\in GL(2,\mathbb{Z})$ such that $\det(g)=1$. Show that $SL(2,\mathbb{Z})$ is non-abelian.

Exercise 5.6.9   Show that if $G$ is a group such that $g^2=1$ for all $g\in G$ then $G$ is abelian.