Groups – Ruficalix

What is the Gist of an “Abelian Group”?

“In mathematics, an abelian group, also called a commutative group, is a group^[1] in which the result of applying the group operation to [any] two group elements does not depend on the order in which they are written. That is, the group operation is commutative^[2]” (“Abelian Group,” Wikipedia, retrieved 12/15/2020, bold added for emphasis). One of the best known Abelian groups is the set of real numbers paired with the binary operation^[3] of addition (ibid).

“A group in which the group operation is not commutative is called a ‘non-abelian group’ or ‘non-commutative group'” (ibid).

The group operation of an Abelian group is often denoted with a plus sign (“+“), as opposed to a multiplication sign (“*”) or no sign (ibid). Abelian groups are also sometimes referred to as abelian groups (note the lowercase “a”) (ibid).

[1] See “What is the Gist of a “Group” (Group Theory)?”
[2] See “What is the Gist of “Commutativity” (Mathematics)?”
[3] See “What is the Gist of a “Binary Operation”?”

Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Commutativity” (Mathematics)?

“In mathematics, a binary operation^[1] is commutative if changing the order of the operands^[1] does not change the result… [Commutativity is often known] as the name of the property that says “3 + 4 = 4 + 3” or “2 × 5 = 5 × 2,” [although it] can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, “3 − 5 ≠ 5 − 3”); such operations are not commutative, and so [can be] referred to as noncommutative operations.” (“Commutative Property,” Wikipedia, retrieved 12/15/2020, bold added for emphasis)

“If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation.” (ibid)

The Wikipedia article also has some delightful example of commutativity and non-commutativity in every day life that could be useful in getting the gist of the idea: “Commutative operations in everyday life” and “Noncommutative operations in daily life” (retrieved 12/15/2020).

Note: commutativity means something different in propositional logic, and commutativity is not the same thing as associativity^[2] (ibid)

[1] See “What is the Gist of a “Binary Operation”?”
[2] See “What is the Gist of “Associativity” (Mathematics)?”

Disclaimer:

What is the Gist of the Different Symbols Used to Denote a Group in Mathematics?

(Note: this post is not going to be as simple as most other posts due to time constraints. Instead, the intention of this post is to provide a starting place to understand the different names, like D4 and C5, for groups)

In the attached doc, I briefly cover the following:

Z (The Integers)
Z/nZ (The Integers Modulo n)
Z_n (The Integers Modulo n (in Group Theory))
C (Cyclic group)
C_n (Cyclic group of degree/order n)
S (Symmetric Group)
S_n (Finite Symmetric Group)
A_n (Alternating Groups)
D_n (Dihedral Group).

Names-of-Groups-Doc Download

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What is the Gist of a “Group” (Group Theory)?

In general, a group can be thought of as a set^[1] that has been combined with a binary operation^[2] and follows certain rules regarding what happens when various elements of the set are combined with each other using that binary operation.

More precisely, “a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms^[3] are satisfied, namely closure,^[4] associativity,^[5] identity^[6] and invertibility.^[7]” (“Group (mathematics),” Wikipedia, retrieved 10/17/2020)

I particularly like Wikipedia‘s example of groups and their definition of groups that follows, so I’ve included the information below:

“Example: the integers

“One of the most familiar groups is the set of integers $\mathbb {Z}$ which consists of the numbers

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …,

together with [the binary operation] addition.

“The following properties of integer addition serve as a model for the group axioms given in the definition below.

“For any two integers a and b, the sum a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition.
“For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.
“If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.
“For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

“The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

Definition

“A group is a set, G, together with [a binary operation “⋅”]… that combines any two elements a and b to form another element, denoted a ⋅ b or ab. [Note that ⋅ is called the group law of G]. To qualify as a group, the set and operation, (G, ⋅), must satisfy four requirements known as the group axioms:

“Closure: For all a, b in G, the result of the operation, a ⋅ b, is also in G.
“Associativity: For all a, b and c in G, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).
“Identity element: There exists an element e in G such that, for every element a in G, the equation e ⋅ a = a ⋅ e = a holds. Such an element is unique… and thus one speaks of the identity element.
“Inverse element: For each a in G, there exists an element b in G, commonly denoted a⁻¹ (or −a, if the operation is denoted “+”), such that a ⋅ b = b ⋅ a = e, where e is the identity element.

…

“The set G is called the underlying set of the group (G, ⋅). Often the group’s underlying set G is used as a short name for the group (G, ⋅). Along the same lines, shorthand expressions such as “a subset of the group G” or “an element of group G” are used when what is actually meant is “a subset of the underlying set G of the group (G, ⋅)” or “an element of the underlying set G of the group (G, ⋅)”. Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.”

Source: “Group (Mathematics),” Wikipedia, retrieved 10/17/2020

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What is the Gist of “Invertibility” (Mathematics)?

Very loosely speaking, the inverse element undoes the effect of combining two elements, and elements combined with their inverse element return the identity element.^[1]

To be more precise, suppose there exists a binary operation^[2] acting on a set^[3]. If an element of this set is put into the binary operation with the element’s inverse, the result is the identity element of the set. Furthermore, if an element is combined with another element of the set via a binary operation, combining the result with the original element’s inverse will return the second element, e.g., 5+6=11, then 11+(-5)=6, where 5 was the original element, (-5) is the inverse of 5 under the addition operation, and 11 was the result of the original combination.

If an element has an inverse, it is considered “invertible.” If every element in a set has an inverse, the set is considered to have the property of invertibility.

Additional points to note:

In the above definition, we assumed the set is “closed”^[4] under the binary operation and that there existed a two-sided identity element in the set.
Generally, no two elements in a set have the same inverse element.
If the order that an element goes into the binary operation matters, there can be elements which behave like an inverse element if put in first but not when put in second. It’s also possible to have elements which behave like an inverse element if put in second but not when put in first.
The inverse element of arbitrary element a is frequently called a^-1.
For examples of inverse elements, see https://en.wikipedia.org/wiki/Inverse_element#Examples.
It’s also possible to have more generalized version of an inverse without an identity. For more information, see this link.

Source: “Inverse Element,” Wikipedia, retrieved 10/25/2020

Disclaimer:

What is the Gist of “Identity” (Mathematics)?

When one of the two inputs of a binary operation^[1] that is acting on a set^[2] is an identity element, the output will be the other element that was used as an input.

Very loosely speaking, the identity element doesn’t change anything when combined with things. E.g., multiplying 3 by 1 (a multiplicative identity) results in 3, or adding 0 (an additive identity) to 5 results in 5.

Additional points to note:

Which element of the set is the identity element depends on which binary operation is acting on the set.
The identity element is often just called “the identity” if it is clear which binary operation and set you are talking about.
If the order that an element goes into the binary operation matters, there can be elements which behave like an identity element if put in first but not when put in second. It’s also possible to have elements which behave like an identity element if put in second but not when put in first, e.g., 5-0=5 but 0-5=-5.
It is only possible for there to be zero or one identity elements which behave like an identity element regardless of the order it is put into the binary operation.
For examples of identity elements, see https://en.wikipedia.org/wiki/Identity_element#Examples

Source: “Identity element,” Wikipedia, retrieved 10/10/2020

[1] See “What is the Gist of a “Binary Operation”?“

[2] See “What is the Gist of a “Set” (Mathematics)?“

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What is the Gist of “Associativity” (Mathematics)?

Some binary operations^[1] have something called “the associative property… which means that rearranging the parentheses in an expression will not change the result.”

To elaborate, let “*” be a binary operation^[1]. The operation “*” is associative when acting on a specific set^[2] S only if the following statement is always true: (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z in S

Note: associativity means something different in propositional logic, and associativity is not the same thing as commutativity^[3].

Source of quotes and information: “Associative property,” Wikipedia, retrieved 10/10/2020. Bold added to quote for emphasis.

[1] See “What is the Gist of a “Binary Operation”?“
[2] See “What is the Gist of a “Set” (Mathematics)?“
[3] See “What is the Gist of “Commutativity” (Mathematics)?“

Disclaimer:

What is the Gist of “Closure” (Mathematics)?

“A set^[1] is closed under an operation if performance of that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction: $1-2$ is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because $0+0=0$ , $0-0=0$ , and $0\times {0}=0$ ).

“Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually” (“Closure (mathematics),” Wikipedia, retrieved 10/9/2020, bold added or taken away for emphasis).

[1] See “What is the Gist of a “Set” (Mathematics)?“

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