Group theory

Group theory is the study of the algebraic structures called groups. A group is a set that, in an abstract sense, has a special kind of "structure" with some very "nice" properties. Many of the sets commonly used in mathematics, like the integers and the complex numbers, are groups.

Group theory provides a basic foundation to study other algebraic structures that have even more structure, like rings and fields.

Concepts from group theory

A group

A group is a set $G$ and a binary operator $\cdot$ that has the following properties:

• The group has an identity element: There is an element $e$, such that $x\cdot e=x$ and $e\cdot x=x$ for all $x$ in the group.
• Every element has an inverse: For each element $x$ in the group, there is another element $y$, such that $x\cdot y=e$ and $y\cdot x=e$. ($e$ is the identity element)
• The operation is associative: For all elements $x$, $y$ and $z$ we have $x\cdot (y\cdot z)=(x\cdot y)\cdot z$.

Notation for groups

A group can have only one identity element, and although this element is generically labeled $e$, it is often relabeled depending on the group being described. Examples of this notation will be shown later, but the identity element may be called $0$ (often for abelian groups), $1$ (usually for multiplicative groups), or $I$ (in groups of matrices).

The inverse of an element gets its own notation, again depending on the context. In multiplicative groups (groups where the operation reminds us of multiplication) the inverse of an element $x$ is written $x^{-1}$ and in additive groups the inverse of $x$ is usually written $-x$.

Subgroups and normal subgroups

A subgroup is a subset of a group that is itself a group. Not every subset of a group is a subgroup (for example, a subset that does not contain the identity element e cannot be a group). A normal subgroup is a very important kind of subgroup and is defined by a few different equivalent definitions. The role of normal subgroups will be shown in the next few sections.

Special kinds of groups

An abelian group is a group with an operation that is commutative. That is, $x\cdot y=y\cdot x$ for every element x and y in the group. Abelian groups are often easier to analyze than non-abelian groups. For example, all subgroups of an abelian group are normal. As a matter of style, the operator on an abelian group is often called "addition" and the identity element called 0. Conversely, non-abelian groups often (but not always) use the multiplication operator $\cdot$ and call the identity element 1. Again, the notation used is a matter of context and preference.

A cyclic group is a group that is generated by a single element. The cyclic group G generated by the element g is the set of all the integral powers of the element g and its inverse. Every cyclic group is abelian, but the converse is not true (see the examples).

A solvable group, or a soluble group, is a group that has a normal series whose quotient groups are all abelian. A simple group is a group that has no non-trivial normal subgroups. One interesting simple group is the alternating subgroup $A_{5}$, which has 60 elements. Simple groups cannot be solvable, and so $A_{5}$ and the symmetric group $S_{5}$ are not solvable. This is one of the first important results to arise from group theory. The fact that $S_{5}$ is not solvable gives a proof that there is no closed form solution to solve a quintic polynomial (recall that the quadratic equation gives the roots for any 2nd-degree polynomial; this result states that there is no such equation for a 5th-degree polynomial, or any other general polynomial with degree larger than 4).

A free group is a group in which every element of the group is a unique product, or string, of elements of some subset of the group. Every group is isomorphic to a quotient group of some free group, so understanding the properties of free groups helps us understand the structure of all groups. Free groups are also used to find the presentation of a group, a useful tool used to completely characterize the structure of a group.