# The Fascination of Groups

> Source: https://ollybritton.com/notes/books/the-fascination-of-groups/ · Updated: 2025-01-17 · Tags: safe-to-post-online, the-fascination-of-groups

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"The Fascination of Groups" is a book about group theory.

## Notes
### 2022-03-07
##### What is the set $\mathbb Z^+$ equivalent to??
$$
\mathbb N
$$

##### What does the notation $\mathbb Z\setminus\\{0\\}$ mean??
Non-zero integers.

### 2022-03-08
##### What common notation is used for a general binary operation??
$$
x * y
$$
##### What is true about an operation on a group??
It has to have exactly one answer.

##### Why is the anagram on the set of words not a unary operation??
Because there's not always an anagram, or there might be multiple anagrams.

##### How could you prove that an operation is not commutative??
Show that, in general

$$
x * y \ne y * x
$$

##### What does it mean for an operation to be closed in a set??
The operation always gives an element in the set.

##### Is subtraction closed in the set $\mathbb{N}$??
No, because you can end up with negative numbers.

### 2022-03-09
##### What does the notation $X \cdot Y$ mean??
Do $Y$, then $X$.

##### What could you consider $XY$ shorthand for when dealing with transformations or operations??
$$
X \cdot Y
$$
##### How could you represent the permutation $ABC \to CAB$ with a matrix on $[A B C]$??
$$
\left(\begin{matrix} 0 \& 1 \& 0 \\\\ 0 \& 0 \& 1 \\\\ 1 \& 0 \& 0 \end{matrix}\right)
$$

### 2022-03-10
##### What does it mean for an operator $*$ to be associative for $a * b * c$??
$$
(a * b) * c = a * (b * c)
$$

for all $a, b, c$ in the set of operands.

##### When can't you write $aba$ as $a^2b$??
When the operation implied is not commutative.

##### When is a fixed element of a set $e$ the RIGHT-identity under an operation $*$??
When

$$
x * e = x
$$

for all $x$ in the set.

##### When is a fixed element of a set $e$ the LEFT-identity under an operation $*$??
When

$$
e * x = x
$$

for all $x$ in the set.

##### Consider an arbitrary element of a set $x$ and a binary operation $*$. How could you write the difference between left-identity $\alpha$ and right-identity $\beta$??
$$
\alpha * x = x
x * \beta = x
$$

##### What is the residue or congruence class of a number $a$ under modulo $n$??
The set of all numbers congruent to $a$ modulo $n$.

##### Can you do a quick proof that if a system has left and right identities $l, r$ then they must be the same??
$$
lr = l = r
$$

##### What is an equivalence class for some $x$??
The set of all things equivalent to $x$ under whatever operation you're working with.

##### What does it mean for a set of congruence classes ${a, b, c...}$ to be a complex residue system modulo $n$??
$$
a \cup b \cup c \cup ... = \mathbb{Z}
$$

### 2022-03-14
##### What does it mean for $y$ to be an inverse of $x$ in a set??
$$
x * y = y * x = e
$$

where $e$ is the identity.

##### Why is not always the case that $y$ is the inverse of $x$ if $$x * y = y * x = e$$??
Because you must specify that there is always an inverse in the set.

##### What is true about left and right inverses in associative systems??
They are equal.

##### Why doesn't subtraction with respect to real numbers have an inverse??
Because the left-inverse isn't always the same as the right-inverse.

##### What is the inverse of the associative operation $x * y$??
$$
y^{-1} * x^{-1}
$$

##### How can you prove $(x * y)^{-1}$ for an associative operation??
Multiply

$$
(x*y)(y^{-1} * x^{-1})
$$

### 2022-03-15
##### A set $G$ has a binary operation $*$. What are the four properties required for this to be a group??
1. C: The set is closed under $*$.
2. A: The operation is associative.
3. N: There is an identity/neutral element.
4. I: Each element has an inverse.

##### What is the wordy definition of a group??
"A group is a mathematical system consisting of elements, each of which has an inverse, which can be combined by some operation without going outside the system."

##### What is the order of a finite group??
The number of elements in the group.

### 2022-03-17
##### What appears first in a group table??
The identity element.

##### If you wanted to look up the operation $X * Y$ on a group table, where would you look for $X$??
Down.

##### If you wanted to look up the operation $X * Y$ on a group table, where would you look for $Y$??
Across.

##### ![PHOTO GROUP TABLE INTERSECTION](group-table-intersection.png) What operation leads to $Y$ here??
$$
Z * X
$$

##### In what order do you do the operation $xy$??
$y$, then $x$.

##### What's another name for a group table??
A Cayley table.

##### What is the name for a commutative group??
An Abelian group.

##### If a group is Abelian, what is true about its Cayley table??
It is symmetric along the diagonal.

##### What does it mean for a group to be Abelian??
The operator is commutative.

##### Why is the set of rotations about arbitrary points in a plane not a group??
Because you can combine rotations to get a translation.

##### What is a symmetry group??
The group of all transformations on an object that leave its appearance unchanged.

### 2022-03-17
##### What is the cancellation law for groups??
$$
xa = ya \implies x = y
$$

$$
ax = ay \implies x = y
$$

##### $$xa = ay \implies x = y$$ Why is this not true in general for groups??
Because groups aren't necessarily commutative/Abelian.

##### What is the Latin square property of a group table??
There are no repeats in rows or columns.

##### Why isn't a Latin square necessarily a group table??
Because associativity may fail.

##### Because of the Latin square property, what is true about each row or column of a group table in relation to the elements of the group??
It is a permutation of the elements.

##### Why is it always possible to "solve" $ax = b$ or $xa = b$ in a group??
Because of the Latin square property, there is always one solution.

##### What is Cayley's theorem??
Any finite group of order $n$ is isomorphic to a subgroup of $S_n$.

##### What is $S_n$ or $P_n$??
The group all permutations of $n$ symbols.

##### What is the order of $S_n$ or $P_n$??
$$
n!
$$

##### What is the symbol for the group of all permutations of $n$ symbols (the "symmetric group")??
$$
S_n \text{or} P_n
$$

##### What's the name for $S_n$ or $P_n$??
The symmetric group.

### 2022-03-18
##### How can you use Cayley's theorem to simplify the process of showing that a Latin square isn't a group table??
Show that the permutations represented by each row aren't closed.

##### How do you arrange the elements in a group table in order to write out the regular representation of finite groups by matrices??
By arranging elements with their corresponding inverses along the column.

##### What does arranging elements with their inverses along the column in a group table mean for the diagonal??
It consists only of the identity.

##### Once you've got $eabcdef...$ along the row and $ea^{-1}b^{-1}c^{-1}d^{-1}e^{-1}f^{-1}$ along the diagonal, what do you do to find the matrix representation for $p$??
Create an $n \times n$ matrix where there is a $1$ if there is a $p$ in that position.

$$e\,p\,q\,a\,b\,c$$ $$q\,e\,p\,c\,a\,b$$ $$p\,q\,e\,b\,c\,a$$ $$a\,c\,b\,e\,q\,p$$ $$b\,a\,c\,p\,e\,q$$ $$c\,b\,a\,q\,p\,e$$
What is the matrix representation for $p$??::

$$
\left(\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}\right)
$$

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