# 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? Note by Krishnan Shankar 4 years, 7 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as *italics* or _italics_ italics **bold** or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote  # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in \( ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123}` $\boxed{123}$

## Comments

Sort by:

Top Newest

thnx..................

helped a lot..................:)

- 4 years, 7 months ago

Log in to reply

The set of all whole numbers with the operation + is a group: the identity element is of course 0.

You mean integers right? Otherwise there would be no inverse element.

- 4 years, 7 months ago

Log in to reply

Of course. I include the negative integers among the whole numbers.

- 4 years, 7 months ago

Log in to reply

Look up dihedral group D3 as an example of a non-abelian group with 6 elements. It's the symmetry group of the equilateral triangle.

Lie groups, an example of a "continuous transformation group", would be an example of an non-abelian group of infinite order. They play a foundational role in differential manifolds.

Group theory is extremely important in theoretical physics, involving both discrete and continuous groups.

- 4 years, 7 months ago

Log in to reply

Non-abelian group with infinitely many elements - They're everywhere; the groups $GL_n$ and $SL_n$ over an "infinite" field would be the best examples.

Group theory arose from the study of symmetry in geometric objects, namely the triangle & the square.

- 1 year, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...