MTH3003 Lecture 3

Simon Smith

Abstraction in mathematics is key.

Basically just a bunch of recapping definitions, all listed below.

Subgroups

A subgroup is defined within a group $G$ with operation $\ast$. We say that $H$ is a subgroup of $G$ if it is a subset of $G$, whilst being a group itself (with the same operation as $H$). The notation we use is $H\le G$, and sometimes say that $G$ is a supergroup of $H$.

We can check this quickly using the Quick Subgroup Test. Given that $H$ is a subset of $G$, where $G$ is a group with operation $\ast$, then $(H,\ast )$ is a subgroup of $G$ if and only if:

1. $H$ contains the identity element, i.e., $e_{G\in H}$.
2. $H$ is closed under $\ast$, i.e., for all $h_1,h_2\in H$ we have $h_1\ast h_2\in H$.
3. Every element of $H$ has an inverse in $H$, i.e., for all $h\in H$ we have $h^{-1}\in H$.

Given this, we can prove the following properties:

1. The identity element in $H$, $e_H$, is the same as the identity element in $G$, $e_G$.
2. For all $h\in H$, the inverse of $h$ in $H$ is equal to the inverse of $h$ in $G$.
3. $|H|=1$ if and only if $H=⟨e_G⟩$ (the trivial group)
4. If $|G|$ is finite, then $|H|=|G|⟺H=G$.

If the subgroup is not equal to the supergroup, then we call it a proper subgroup. Hence, proper nontrivial subgroups are what we'll mostly be using.

Cyclic Groups

A cyclic group is defined by a group $G$ with operation $\ast$. We say that $G$ is a cyclic group if it can be generated by a generator element $g\in G:g^n=\underset{n\text{ times}}{\underbrace{g\ast g\ast \cdots \ast g}}$, where $g^0=e_G$ and $g^{-n}=(g^n{)}^{-1}=(g^{-1}{)}^n$. We write this as $⟨g⟩\le G$, such that $|⟨g⟩|=o(g)$.

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Pre-Lecture Notes from University Notes

• Subgroups - bunch of definitions and theorems.
• Cyclic groups - bunch of definitions and theorems.
• Vaguely recap, so not doing too much beforehand.