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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and a binary operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot} that has the following properties:

• The group has an identity element: There is an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \cdot e = x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \cdot x = x} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the group.
• Every element has an inverse: For each element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the group, there is another element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \cdot y = e} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \cdot x = e} . (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is the identity element)
• The operation is associative: For all elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} (often for abelian groups), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} (usually for multiplicative groups), or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{-1}} and in additive groups the inverse of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is usually written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_5} , which has 60 elements. Simple groups cannot be solvable, and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_5} and the symmetric group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_5} are not solvable. This is one of the first important results to arise from group theory. The fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Examples of groups

Abelian examples

The integers together with addition is a cyclic group. Consider the required properties:

• Addition is a well-defined binary operator. Whenever a and b are integers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+b} is also an integer, and so the operation is closed on the group.
• The element 0 is the additive identity for the group. It is easy to see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+0=a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0+a=a} .
• Every element has an inverse. For example, the inverse of 2 is -2, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2+(-2)=0} .
• Addition is associative. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b)+c=a+(b+c)}
• The group is abelian. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a+b=b+a}
• The group is cyclic with 1 as its generator. For example, 2=1+1, 3=1+1+1, 4=1+1+1+1, and so on. Further, -3=(-1)+(-1)+(-1), the third multiple of the inverse of 1.

By similar reasoning, the real numbers with addition is an abelian group, but it is not cyclic. There is no single element such that every real number is some integral multiple of that element (if you suppose that there is such an element, say a, then a/2 is not an integral multiple of a)

The integers with multiplication, however, is not a group. Multiplication is well-defined and associative, and 1 is an identity element. However, not all elements have inverses. For example, the inverse of 2 should be 1/2 (since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\cdot\frac{1}{2}=1} ), but 1/2 is not an integer.

However, the real numbers with multiplication are still not a group. It is almost a group, since all the nonzero real numbers have an inverse (the inverse of a is 1/a). But, zero has no inverse. This single failure means that the real numbers with multiplication are not a group.

Nonabelian examples

One of the first examples of non-commutative multiplication arises in linear algebra. Matrix multiplication is not commutative (the product AB is not the same as BA).

The set of all n-by-n invertible real matrices together with matrix multiplication is a group. This group is called the general linear group of degree n, and is written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GL_n(\Bbb{R})} .

As a finite example, the symmetric groups of degree greater than 2 are all nonabelian.

Operations involving groups

Morphisms

A homomorphism is a map from one group to another group that preserves the multiplicative structure of the groups; written formally, the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi:G \rarr H} obeys the rule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(a \cdot b) = \phi(a) \cdot \phi(b)} . The kernel of a homomorphism is the set of all elements that map to the identity element; this set is a normal subgroup.

An isomorphism is an injective homomorphism (or, equivalently, one whose kernel consists only of the identity element). We say that the two groups are isomorphic. Isomorphic groups can be thought of as being the same group with their elements relabeled, since their structures are identical. Isomorphisms are important because they allow us to think of the same group in different ways. Take a certain group, and we could consider its isomorphic version in the symmetric groups (from Cayley's theorem), its isomorphic version as a quotient group of a free group, or a group of matrices over some field (as a group representation).

An automorphism is an isomorphism of a group onto itself. The set of all automorphisms for a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , often written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Aut(G)} , is itself a group! One important subgroup of the automorphism group is the set of all inner automorphisms. An inner automorphism is a conjugation mapping. For an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} , the inner automorphism corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is the mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_g: x \mapsto gxg^{-1}} . The inner automorphism subgroup is isomorphic to the quotient group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/Z(G)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z(G)} is the center of the group. The inner automorphism subgroup is normal inside Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Aug(G)} , and the quotient group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Aut(G)/Inn(G)} is called the outer automorphism group. Finding non-inner automorphisms for a group is, in general, difficult. Because abelian groups have a trivial inner automorphism subgroup, finding automorphisms for abelian groups is, strangely enough, harder than for nonabelian groups.