What is a group?

One of the first abstract notions a student encounters after basic calculus and linear algebra is that of a group. It is a fundamental object in mathematics which shows up in chemistry, cosmology and even in a hidden way in music (although we won't explore these here in this note).

So what is a group? A group is a set with a binary operation that satisfies four conditions. A binary operation is simply a way of taking two elements of the set and combining them to yield a third element. So what conditions make it a group? Let us call the set $G$ and the the operation as $\ast$. The set $G$ could be finite or infinite (which leads to finite group theory and infinite group theory). Then $(G, \ast)$ must satisfy: (i) The operation $\ast$ is closed i.e., given elements $g_1, g_2 \in G$ the composition $g_1 \ast g_2$ must also lie in $G$; (ii) the set $G$ has a distinguished element called the identity, say $e$ such that for all $g \in G$ we have $g\ast e = e\ast g = g$; (iii) Every element $g \in G$ has an `opposite' i.e., an inverse, say $h \in G$ such that $g \ast h = h \ast g = e$, the identity; (iv) The operation $\ast$ is associative i.e., for all $g, h, k \in G$ we have \((g\ast h)\ast k = g \ast (h\ast k). This last condition is to ensure that collections of self maps of mathematical objects form groups.

Note that a group is not required to be commutative; it may happen for two given elements \(g, h \in \) that $g\ast h \neq h \ast g$. If on the other hand $g\ast h = h\ast g$ for all possible pairs of elements, then the group is said to be abelian (in honor of Niels Henrik Abel who showed that the quintic polynomial equation is not solvable by a formula).

Some examples of groups: The set of all whole numbers (including all negative integers) with the operation $+$ is a group: the identity element is of course 0. The set of all positive real numbers under multiplication $\cdot$ is a group with identity element 1. But here is a much more interesting example discovered in 1843 by the great Irish mathematician William Rowan Hamilton (he was so excited by this discovery that he carved it into Brougham Bridge over the Royal Canal in Dublin.) Here it is:

$G = \{1, -1, i, -i, j, -j, k, -k\}$ with operation multiplication.

has 8 elements with the special elements $i, j, k$ which satisfy the following conditions:

$i^2 = j^2 = k^2 = ijk = -1, \quad ij = k, \quad jk = i, \quad ki =j$

So in this group what is $ji$? Let's see, $ij =k$ so $ijij = k^2 = -1$. Therefore, $i (ji) j = -1$ so $(-i) i (ji) j (-j) = (-i) (-1) (-j)$. The left side then simplifies to $ji$ while the right side becomes $- ij = -k$. Hence, note that $ij \neq ji$ and we have a non-abelian group. This famous group is called the group of quaternions and is usually denoted as $Q_8$. You can think of $i, j, k$ as three different flavors of $\sqrt{-1}$ but if that makes you uncomfortable there is also a realization of this group in terms of matrices:

$1 = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}, \quad -1 = \begin{pmatrix} -1 & 0\\0 & -1\end{pmatrix},$

$i = \begin{pmatrix} 0 & 1\\-1 & 0 \end{pmatrix}, \quad j = \begin{pmatrix} \sqrt{-1} & 0\\ 0 & -\sqrt{-1}\end{pmatrix},$

$k = ij = \begin{pmatrix} 0 & -\sqrt{-1}\\-\sqrt{-1} & 0\end{pmatrix}$

This is just the beginning of a vast and beautiful area of mathematics. So reader can you find a non-abelian group with 6 elements? How about a non-abelian group with infinitely many elements?