Groups, Subgroups and Isomorphisms

The three most commonly studied algebraic structures are groups, rings, and fields. Groups can be found in many areas of mathematics such as geometry, analysis, topology, and other fields as well. We will continue with formal definitions.

A group is a set \(G\) with one binary operation that adheres to the following (using multiplication or juxtaposition for this binary operation):

1. The operation is associative. So, for all \(g_{1}, g_{2}, g_{3} \in G\), \((g_{1}g_{2})g_{3} = g_{1}(g_{2}g_{3})\).
2. There exists an identity for the operation. So, for all \(g \in G\), an element \(1 \in G\) such that \(1g = g\) and \(g1 = g.\)
3. Every \(g \in G\) has an inverse for the operation. So, for all \(g \in G\) there exists \(g^{-1}\) with the property that \(gg^{-1}=1\) and \(g^{-1}g=1\)

If the operation is commutative along with 1, 2, and 3 above, then the group is an abelian group. The order of group \(G\) is the number of elements in \(G\), denoted as \(|G|\). If \(|G| > \infty\), then \(G\) is an infinite group if, othewise \(G\) is a finite group.

The integers \(\mathbb{Z}\), rationals \(\mathbb{Q}\), and reals \(\mathbb{R}\) all form groups under addition and are all albenian groups. The elements of the set of rational numbers \(\mathbb{Q}\) are not a group under multiplication because \(0\) does not have a multiplicative inverse. For a set to be closed under multiplication, any element \(x\) must have a multiplicative inverse \(-x\). The elemenet \(0 \in \mathbb{Q}\) under multpication \(\times\) does not have an inverse. However, the set of nonzero elements of \(\mathbb{Q}\) under multiplication is an abelian group. This is because the nonzero elements of \(\mathbb{Q}\) with one binary operation satisfies 1, 2, 3 above and is commutative under multiplication.

Rings and the Rings of Integers

A ring, formally defined, is a set \(R\) with two binary operations defined in the set. The two binary operations are normally called addition \(+\) and multiplication denoted as \(\cdot\) or by juxtaposition. The binary operations satisfy the following six axioms: (Note: the operations "addition" and "multiplication" may be very different from the usual addition and mutiplication of numbers)

1. Addition is commutative. Meaning, \(a+b=b+a\) for every pair \(a,b\in R\).
2. Addition is associative. Meaning, \(a+(b+c)=(a+b)+c\) for \(a,b,c\in R\).
3. There exists an additive identity, denoted by 0. Meaning, that \(a+b=a\) for every \(a\in R\).
4. For every \(a\in R\) there exsists some additive inverse denoted by \(-a\) such that \(a+(-a)=0\).
5. Multiplication is associative. Meaning, \(a(bc)=(ab)c\) for \(a;b;c\in R\).
6. Multiplication is left and right distributive and right distributive over addition. Meaning, \(a(b+c)=ab+ac\) and \((b+c)a=ba+ca\) for for \(a;b;c\in R\).

Further, if the binary operations on set \(R\) also satisfy the following axiom:


7. Multiplication is commutative then \(R\) is a commutative ring. Meaning, \(ab=ba\) for every pair \(a,b\in R\).

Further, if for the binary operations on set \(R\):


8. There exsists a multiplicative identity denoted by \(1\) such that \(a\cdot 1=a\) and \(1\cdot a=a\) for every \(a\in R\) then \(R\) is a ring with identity. If \(R\) satisfies axioms 1 through 6 and 8 but not 7, then \(R\) is a ring with unity.
9. Satify axioms 1 through 8 then \(R\) is a commutative ring with an identity or a unity.

Recall that a set \(G\) with one operation, say \(+\), defined on it and satisfies axioms 1 through 4 is called an abelian group. This means each ring is an abelian group, since, relative to the operration \(+\) axioms 1 through 4 for a ring must be satisified by definition.

Looking now at the basic number number systems \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\), are examples of rings, both commutative and noncommutative. We will see that the ring algebraic structure is prevalent in mathematics.

For some definitions, a ring \(R\) with only one elmenet is called trivial. The ring is trivial if and only if \(0=1\) according to the binary oprations in \(R\). A finite ring is a ring \(R\) with a finite number of elements, otherwise \(R\) is an infinite ring. Looking back at our basic number systems, \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) are all infinite rings.

To continue.

Vector Spaces Over a Field

Binary Operations

Binary Operations take two elments of a set and produce a new element that is a part of the same set. Binary Operations are fundemetal to Algebrai structures. Let \(S\) be a nonempty set. A binary operation on set \(S\) is a function from \(S \times S\) to \(S\). We can denote the image of the ordered pair \((a, b) \in S \times S\). We can poorly denote the binary opration by this image as \(ab \in S\).

Put differently, a binary operation on a set \(S\) is given when every ordered pair (\(a,b)\) of elements of set \(S\) is an associated unique element \(c \in S\). A binary operation on on set \(S\) must be closed if \(c \in S\).