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.