The Fascination of Groups, Budden
“The Fascination of Groups” is a book about group theory.
Things I Don’t Understand
- Exercises 3:
- Q7: “Examine numbers of the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$ for closure under addition, subtraction, multiplication and division for integers and rationals”
- Answers say it is closed for multiplication for integers and multiplication and division for rationals. How is this possible? Consider the case where $b, c = 0$ and $a$ is $1$. For division, this would be the same as dividing by $\sqrt{2}$, which could eliminate the square root and mean the product wasn’t in the set.
- Q14: About compositions of functions of a certain kind being closed. I may have substituted wrong but I can get a quadratic?
- Q20: Don’t understand the operation.
- Q7: “Examine numbers of the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$ for closure under addition, subtraction, multiplication and division for integers and rationals”
Notes
2022-03-07
What is the set $Z^+$ equivalent to?
What does the notation $Z\0$ mean?
Non-zero integers.
2022-03-08
What common notation is used for a general binary operation?
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?
How could you represent the permutation $ABC \to CAB$ with a matrix on $[A B C]$?
2022-03-10
What does it mean for an operator $*$ to be associative for $a * b * c$?
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 $\beta$?
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?
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 $n$?
2022-03-14
What does it mean for $y$ to be an inverse of $x$ in a set?
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 $x * y$?
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?
- C: The set is closed under $*$.
- A: The operation is associative.
- N: There is an identity/neutral element.
- 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.
What operation leads to $Y$ here?
What operation leads to $Y$ here?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?
\[xa = ay \implies x = y\]
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 $P _ n$?
What is the symbol for the group of all permutations of $n$ symbols (the “symmetric group”)?
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.