Group theory

In mathematics, groups often arise as structures representing the set of possible symmetries of of some object. We have been intentionally vague about the meaning of the terms symmetry and object. The terms may have an obvious geometric sense. For example, the set of orientation-preserving continuous transformations of the Euclidean plane $\mathbb {R} ^{2}$ that take lines to lines and preserve angles can be decomposed into tranformations of two types: rotations which rotate the whole plane around the origin by an angle of $\theta$, and translations, which shift the origin to a new point and move the whole plane along with it without any "twisting". In coordinates, we can write an arbitrary rotation as

$(x,y)\mapsto (x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta )$,

and translations as

$(x,y)\mapsto (x+x_{0},y+y_{0})$.

Of course, rotations and translations can be combined, and the full group of symmetries is called the 2-dimensional affine group. Felix Klein pioneered this approach to geometry, calling it the Erlangen Program. It would be subsequently taken up and generalized by Élie Cartan and remains an important approach to differential geometry.

But it would be misleading to focus entirely on geometric examples. In fact, the very word group was introduced by Évariste Galois in the context of algebraic equations, as part of what is now called Galois theory. Consider the polynomial $x^{2}+1$, it is irreducible over the real numbers, meaning that we cannot find two real numbers $a$ and $b$, such that

$x^{2}+1=(x-a)(x-b)$.

However, over the complex numbers we can write

$x^{2}+1=(x+i)(x-i)$.

Now, complex conjugation, that is, the mapping $a+bi\mapsto a-bi$, can be thought of as a "symmetry" of the complex numbers. Now, what happens if we take the complex conjugate of the constant values in $(x+i)(x-i)$? We get, $(x-i)(x+i)$; that is, the order of the factors is reversed, but when we multiply it out, the product is still $x^{2}+1$. It turns out that complex conjugation and doing nothing are the two "symmetries" of the complex numbers having thd property that a) the ordinary algebraic operations of addition, multiplication, taking reciprocals and multiplying by a real number and b) the polynomial $x^{2}+1$ are left unchanged. This two element group (written $\mathbb {Z} _{2}$) is called the Galois group of the extension $\mathbb {C} /\mathbb {R}$ or, more simply, of the polynomial $x^{2}+1$. This is but one example of how groups can arise in mathematics where we are not directly concerned with symmetries of geometric objects.

Definitions of groups and subgroups

Groups

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 $x\cdot (y\cdot z)=(x\cdot y)\cdot z$.

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 there is a subset such that every element of the group can be writtenly uniquely as the product, or string, of elements of the subset. 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, $a+b$ 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 $a+0=a$ and $0+a=a$.
• Every element has an inverse. For example, the inverse of 2 is -2, because $2+(-2)=0$.
• Addition is associative. $(a+b)+c=a+(b+c)$
• The group is abelian. $a+b=b+a$
• 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 $2\cdot {\frac {1}{2}}=1$), 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 $GL_{n}({\mathbb {R}})$.

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 $\phi :G\rightarrow H$ obeys the rule $\phi (a\cdot b)=\phi (a)\cdot \phi (b)$. 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 if there is a surjective (thus bijective) isomorphism between them. Isomorphic groups have identical structure and are often thought of as just being relabelings of one another. Isomorphisms are important because they allow us to think of the same group in different ways. For example, 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 as 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 $G$, often written $Aut(G)$, 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 $g\in G$, the mapping $T_{g}:x\mapsto gxg^{-1}$ is an inner automorphism). The inner automorphism subgroup is isomorphic to the quotient group $G/Z(G)$, where $Z(G)$ is the center of the group. The inner automorphism subgroup is normal inside $Aug(G)$, and the quotient group $Aut(G)/Inn(G)$ 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.

Group actions

A group action is a mapping of each of the elements of a group to a bijective mapping on a set. Group actions are incredibly useful. For example, the Sylow theorems are easily proved by considering a group acting on the set of its maximal p-subgroups. Group actions also give rise to the so-called orbit-stabilizer theorem, a very powerful counting theorem. As an application of this, it is easy to show that every finite p-group must have a non-trivial center by considering how a p-group acts on itself via conjugation.

Constructing groups from other groups

There are a few tools in group theory that give us ways to construct new groups from other groups.

A quotient group $G/N$ is the set (and group) of cosets of a normal subgroup $N$ of a group $G$. There is a canonical homomorphism from $G$ onto the quotient group (for which $N$ is the kernel), and so multiplication in the quotient group is related to multiplication in the original group. However, it is important to note that the quotient group is an entirely new group. There is no reason to expect (in general) that there is a subgroup of $G$ that is isomorphic to the quotient group.

The direct product is another way to construct new groups. In the case of two groups $G$ and $H$, the direct product $G\times H$ is the set of all tuples $(g,h)$ where each component comes from $G$ and $H$, respectively. Multiplication is done component-wise. Direct products result in groups that are "larger", in the sense that there is an isomorphic copy of both $G$ and $H$ in the direct product. When the groups are abelian, it is common to call the direct product as the direct sum.

The direct product of more than two groups is certainly possible, and the concept generalizes easily to any countable set of groups. However, in the case of countably-infinite number of groups, there is an important distinction between direct sum and direct product. The tuples in a countably-infinite direct sum are restricted to those that have a finite number of non-identity entries. Direct products have no such restriction.

The semidirect product is another generalization of the direct product in which the multiplication is not done independently in the components. The semidirect product $N\times _{\phi }H$ is the set of tuples $(n,h)$ with components from two groups $N$ and $H$, together with a mapping $\phi :H\rightarrow Aut(N)$. This mapping distorts multiplication in the $N$ coordinates, giving rise to a richer set of group constructions.

See also

Ring (mathematics)