Algebraic Structures Week 2

Recap

Definition-of-a-Group

We can always assume that additive integers and multiplicative integers are a group, and hence are associative, too.

Another common examples is $\mathbb{Q}$ and $\mathbb{R}$ over multiplication, which isn’t a group, but ${\mathbb{Q}}^{\ast }$ and ${\mathbb{R}}^{\ast }$ is (these new sets exclude $0$, which causes probably as it’s non-invertible).

What is an Abelian group?

Let $G$ be a group. Then $G$ is called abelian whenever $ab=ba$ for all $a,b\in G$. Basically, a group that obeys the four axioms of a group, as well as commutativity.

Are identity and inverse elements unique?

If we define an operation $\ast$ on ${\mathbb{R}}^{\ast }=\mathbb{R}\{0\}$ by $x\ast y=|x|y$, then there are two elements which do nothing: $1\ast x$ and $-1\ast x$, because the absolute value. Which is the inverse?

The issue here is that this doesn’t define a group, and in fact leads to contradictions such as $-1=1$. Hence, we define a theorem…

Definition-of-Identity-Uniqueness

This can be proven by assuming the negative, that $G$ has two identity elements. Following this, we create a proof by contradiction which then leads to the two identity elements being equal to each other.

The inverse is more logical, where the inverse of an inverse is the original.

Algebraic Structures Practical 1