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


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.


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



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 - Posted by | Algebra, Group Theory | , , ,


  1. […] you’ll recall from our last post we crudely laid out what it meant for a subgroup to be generated by a set. A cyclic group is […]

    Pingback by Review of Group Theory: Cyclic Groups and Cyclic Subgroups (Pt. I) « Abstract Nonsense | December 27, 2010 | Reply

  2. […] noting that we may conclude from a previous theorem that […]

    Pingback by Review of Group Theory: Homomorphisms « Abstract Nonsense | December 28, 2010 | Reply

  3. […] since we may conclude by previous theorem that […]

    Pingback by Review of Group Theory: Cosets and Lagrange’s Theorem « Abstract Nonsense | December 29, 2010 | Reply

  4. […] be any group. We define the map given by to be the commutator bracket. Then the subgroup generated by the set is called the commutator subgroup of and is denoted . Our first claim is that is a […]

    Pingback by Review of Group Theory: The Commutator Subgroup and the Abelianization of a Group « Abstract Nonsense | February 27, 2011 | Reply

  5. […] We say that a group is -generatable if there exists such that generates. Clearly every finite group is -generatable since . We induct on . For this is trivial since by […]

    Pingback by Review of Group Theory: The Structure Theorem For Finite Abelian Groups (Pt. I) « Abstract Nonsense | April 16, 2011 | Reply

  6. […] notion of a group started out simply. Namely, we wanted to describe some kind of set with a function where the […]

    Pingback by Basic Definitions of Rings « Abstract Nonsense | June 15, 2011 | Reply

  7. […] Since each we know from group theory that . Thus, it suffices to prove that is closed under multiplication. That […]

    Pingback by Subrings « Abstract Nonsense | June 15, 2011 | Reply

  8. […] denotes the order in the abelian group . More explicitly, we define to be […]

    Pingback by Characteristic of a Ring (Pt. I) « Abstract Nonsense | June 19, 2011 | Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: