# Abstract Nonsense

## Review of Groups: Definitions and Basics

Point of post: In this post we will review the very basic definitions of group, subgroup, etc. This isn’t supposed to be a full-fledged discussion of group theory, since it’s assumed the reader will have already taken a course in algebra. This is just a quick refresher and something to get notations in sync. It covers just the basic definitions of groups and subgroups along with a few perfunctory theorems

Groups

A group is formally an ordered pair $(G,\star)$ where $G$ is a set and $\star$ is a mapping $G\times G\to G$ which satisfies the following axioms:

\begin{aligned}&\textbf{Grp 1}\quad \star\left(g_1,\star\left(g_2,g_3\right)\right)=\star\left(\star(g_1,g_2),g_3\right)\textit{ for all }g_1,g_2,g_3\in G\\ &\textbf{Grp 2}\quad \textit{There exists some }e\in G\textit{ such that }\star(g,e)=\star(e,g)=e\textit{ for all }g\in G\\ &\textbf{Grp 3}\quad\textit{For all }g\in G\textit{ there exists some }h\in\textit{ such that }\star(g,h)=\star(h,g)=e\end{aligned}

A group $(G,\star)$ is called abelian if in addition to the above axioms it satisfies $\star(g,g')=\star(g',g)$ for all $g,g'\in G$.

It is common to forgo the formalities of above and denote a group by just its underlying set $G$ and denote its group operation by concatenation or, if $G$ is abelian by addition. The element $e$ denoted in $\textbf{Grp 2}$ is known as the identity element and is denoted as $0$ if $G$ is abelian. The element $h$ given in $\textbf{Grp 3}$ is known as the inverse of $g$ and is denoted $g^{-1}$ and $-g$ if $G$ is abelian . A group containing one element (the identity element) is called trivial. The cardinality $\#(G)$ is called the order of $G$ and is denoted $|G|$. If $|G|<\infty$ we call $G$ a finite group. For $g\in G$ we may define arbitrary products of $g$ exponentially by $g^0=e,g^1=g$ and $g^{n+1}=gg^n$, and additively $0g=0,1g=g$ and $(n+1)g=ng+g$. Similarly, we may define $g^{-n}$ as $(g^{-1})^n$ and $-ng=n(-g)$.

If $g\in G$ and the set $K=\left\{n\in\mathbb{N}:g^n=e\right\}$ is non-empty we call $\min K$ the order of $g$ and denote this $|g|$ or $\text{ord}(g)$ when it’s necessary to distinguish between the order of a group and the order of an element. If $K=\varnothing$ then we say that $g$ is of infinite order and denote this by writing $|g|=\infty$.

Some common theorems which I will state but not prove (they can be found in any book on algebra) are

Theorem: The identity element of a group is unique.

Theorem: The inverse of an element of a group is unique.

Theorem: The cancellation law holds, in the sense that if $G$ is a group and $g_1,g_2,g_3\in G$ then $g_1g_3=g_2g_3$ implies $g_1=g_2$.

Theorem (Generalized Associativity): Said simply, any arbitrary placement of parentheses does not change the value of a product of group elements. So, for example $(g_1(g_2g_3(g_4))=g_1(g_2(g_3g_4))$ and so we may write an arbitrary product without parentheses and not be ambiguous.

Subgroups

Let $G$ be a group and $H\subseteq G$. Then, if $H$ is a group under the same binary operation as $G$ then we call $H$ a subgroup of $G$ and denote this by saying that $H\leqslant G$. We call $H$ a proper subgroup if $H\leqslant G$ and $H\ne G$ and denote this $H. We call $H$ a trivial subgroup if it’s trivial. Thus, the non-trivial proper subgroups of $G$ are the $H\leqslant G$ such that $H\ne\{e\},G$.

We prove a few basic facts about subgroups

Theorem: Let $G$ be a group and $\varnothing\subsetneq H\subseteq G$. Then, $\displaystyle \bigcap_{\alpha\in\mathcal{A}}H_{\alpha}\leqslant G$.

Proof: Evidently since $H\ne \varnothing$ we have that there is some $h\in H$ and so by assumption $hh^{-1}=e\in H$. We note then that if $h\in H$ then $h^{-1}e=h^{-1}\in H$. Lastly, we notice that the associativity of the multiplication is inherited from $G$.

The converse is clear. $\blacksquare$

Theorem: Let $G$ be a group and $\left\{H_{\alpha}\right\}_{\alpha\in\mathcal{A}}$ a class of subgroups. Then, $\displaystyle \bigcap_{\alpha\in\mathcal{A}}H_{\alpha}\leqslant G$

Proof: We note that clearly $\displaystyle \bigcap_{\alpha\in\mathcal{A}}H_\alpha\ne\varnothing$ since $e\in H_{\alpha}$ for every $\alpha\in \mathcal{A}$. We note next then that if $\displaystyle h_1,h_2\in\bigcap_{\alpha\in\mathcal{A}}H_{\alpha}$ then $h_1,h_2\in H_{\alpha}$ for every $\alpha\in\mathcal{A}$ and thus $h_1h_2^{-1}\in H_{\alpha}$ for every $\alpha\in\mathcal{A}$ and so $\displaystyle h_1h_2^{-1}\in\bigcap_{\alpha\in\mathcal{A}}H_{\alpha}$ from where the conclusion follows by the previous theorem. $\blacksquare$

Corollary: Let $G$ be a group and $S\subseteq G$. There exists a unique subgroup $\left\langle S\right\rangle$ of $G$ which contains $S$ and is contained in any subgroup containing $S$. In essence, there is a ‘smallest’ subgroup containing $S$.

Proof: Merely take $\displaystyle \left\langle S\right\rangle=\bigcap\left\{H\leqslant G:S\subseteq H\right\}$.

$\langle S\rangle$ is called the subgroup generated by $S$.

From this we can get the following theorem which takes the almost intractable definition of the above idea of generated groups and shows how simple it really is.

Theorem: Let $G$ be a group and $S\subseteq G$, then

$\left\langle S\right\rangle=\left\{x_1\cdots x_n:x_j\in S\text{ or }x_j^{-1}\in S\text{ }j\in[n]\right\}$

Proof: This is clear since evidently the right hand side of the above is a subgroup of $G$ containing $S$ and any subgroup of $G$ containing $S$ must contain the right hand side of the above. $\blacksquare$

References:

1.  Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

December 24, 2010 -

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