Abstract Nonsense

Crushing one theorem at a time

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<G. 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.

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December 24, 2010 - Posted by | Algebra, Group Theory | , , ,

8 Comments »

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