Basic Definitions in Group Theory

Over the past month, I have been reading the first chapter of Dummit and Foote’s Abstract Algebra on basic group theory.¹ The following is a summary of my notes on the first section, detailing basic definitions in group theory.

A binary operation is a map \(*: G\times G \to G\) and for any \(a, b \in G\) we denote \(*(a, b)\) as \(a*b\).

A binary operation is associative if \(a*(b*c)=(a*b)*c\) for all \(a, b, c \in G\), and commutative if \(a*b=b*a\) for all \(a, b \in G\), but generally, binary operations need not be either.

A subset \(H \subseteq G\) is closed under \(*\) if for all \(a, b \in H\), \(a*b \in H\).

Then, a group is an ordered pair \( (G, *) \) where \(G\) is a set and \(*\) is an operation on \(G\) satisfying the following axioms:

1. \(*\) is associative. The generalized associative law, which generalizes associativity to any number of terms, can be proved by induction using the associative law, thus trivializing bracketing within groups.
2. There exists an identity \(e \in G\) such that for any element \(a \in G\), \(a*e = e*a = a\). It is then clear that \(e\) is unique.
3. For each element \(a \in G\) there exists an inverse element \(a^{-1} \in G\) such that \(a*a^{-1} = a^{-1}*a = e\). With a bit more scrutiny, one can see that the inverse is unique as well.

Furthermore, if \(*\) is associative, then \( (G, *) \) is Abelian.

Less formally, we will usually refer to group \( (G, \cdot) \) as \(G\) alone and write \(x\cdot y\) simply as \(xy\) after we define the binary operation \(\cdot\), and we denote the identity of \(G\) as \(1\), and for a positive integer \(n\), we denote \(a^n\) for the product of \(n\) successive terms of \(a\), and \(a^{-n}\) for the product of \(n\) successive terms of \(a^{-1}\). Similar appropriate notation will be defined for additive groups using the \(+\) operator for a natural, familiar system of notation.

Let \(x \in G\), then the order of \(x\), denoted \(|x|\), is the smallest positive integer \(n\) for which \(x^n = 1\). If no such integer exists, then we write \(|x| = \infty\).

A finite group \(G\) is a group where the set \(G\) is finite and we could write \[G = \{g_1, g_2, g_3, \ldots, g_n\}\] Then, the Multiplication Table (also known as the Cayley Table) of \(G\) is the \(n\times n\) matrix \([a_{ij}]\) where \[a_{ij} = g_i g_j \in G\] for all \(i = 1, \ldots, n\) and \(j = 1, \ldots, n\). It contains all the information about a finite group; however, it is computationally inefficient and cannot tell us much about a group’s internal structure.²

1. David Steven Dummit and Richard M. Foote, Abstract Algebra, 3rd ed. (Danvers, MA: John Wiley & Sons, 2004), 13-23. ↩

2. This gets a little more complicated when dealing with infinite groups. ↩