The most familiar group is our number system; we have a bunch of numbers, and we have this thing called "$+$" which can take two numbers and churn out another number. A group is just a set of objects, and you have one "operation" (usually called "$\ast$" or "$+$", but those are just names) that tells you how to combine objects to produce other objects in that set, subject to a few rules. Why devise such a "strange chasing game" in a set of objects? That's because such systems are omnipresent in a stupendous array of phenomena:

- Most
**number systems**(whole numbers, fractions, real numbers, complex numbers...) **"Weird" number systems**that can have strange rules like $1 + 1 = 0$ are used in binary computation, and are crucial in encrypting information for secrecy. Similarly, the "*clock algebra*" has $2359 + 0001 = 0000$.

**Transformation groups**- transformations of space, like a rotation $A$ or a reflection $B$ etc, can be combined into new transformations by defining $B \ast A$ to mean "do $A$ then do $B$". This is absolutely fundamental in physics, from the physical laws of "our space" to investigating the laws of "new spaces", like Quantum Mechanics and Relativity were; and in chemistry, from studying crystals to calculating molecular orbitals. Check "Noether's Theorem" for more.

- The
**Rubik's Cube Group**- every possible configuration is a combination of the basic moves (read the moves from right to left). In the figure below,

- $\mathrm{X1} = \mathsf{NoMove}$
- $\mathrm{X2} = \mathsf{BackTwist}$
- $\mathrm{X3} = \mathsf{TopTwist} \ast\mathsf{BackTwist}$ (read from right to left, so Back Twist then Top Twist)
- $\mathrm{X4} = \mathsf{LeftTwist} \ast\mathsf{TopTwist} \ast\mathsf{BackTwist}$

*I know that each twist can go in two opposing directions, but you get the point.*

- Pretty much every "
**algebraic system**" is built on special kinds of groups - e.g. groups of vectors, groups of matrices, groups of polynomials...

\begin{align}

\begin{bmatrix}

1\\2\\3

\end{bmatrix}+\begin{bmatrix}

4\\5\\6

\end{bmatrix}&&

\begin{bmatrix}

1&0&0\\0&1&0\\0&0&1

\end{bmatrix}+

\begin{bmatrix}

0&1&0\\0&0&1\\1&0&0

\end{bmatrix}&&

\mathsf{Quadratic} + \mathsf{Cubic}

\end{align}

- Many, many, many more...

This exemplifies one power of Mathematics, where a myriad of disparate phenomena and concepts can have a single simple abstract underpinning. Group Theory is the study of that underpinning, thus its results and implications reach far and wide.

## Technical Note

**Associativity.**Any objects $a$, $b$ and $c$ in $G$ must satisfy $(a \ast b) \ast c = a \ast (b \ast c)$. You can verify that this works in the above examples—another "strange" abstract underpinning!**Identity.**$G$ must contain a special object $e$ called the "identity" so that $e \ast a = a = a \ast e$ for every object $a$ in $G$. When adding numbers, $e = 0$ so $0 + a = a = a + 0$. For the Rubik's Cube, $e = \mathsf{NoMove}$ doesn't change the cube, and performing $\mathsf{TopTwist}$ before or after a $\mathsf{NoMove}$ is the same as doing just the $\mathsf{TopTwist}$.**Inverse.**Each object $a$ in $G$ must correspond to an "inverse" $b$ so that $a \ast b = e = b \ast a$. When adding numbers, the inverse of $a$ is $-a$ so that $a + (-a) = 0 = (-a) + a$. For the Rubik's Cube, the inverse of each twist is the same twist but in the opposite direction; $\mathsf{TopTwist} \ast \mathsf{TopOppositeTwist} = \mathsf{NoMove}$ $=$ $\mathsf{TopOppositeTwist} \ast \mathsf{TopTwist}$.