# Elementary group theory

In mathematics and abstract algebra, a group is the algebraic structure , where is a non-empty set and denotes a binary operation called the *group operation*. The notation is normally shortened to the infix notation , or even to .

A group must obey the following rules (or axioms). Let be arbitrary elements of . Then:

**A1**, Closure. . This axiom is often omitted because a binary operation is closed by definition.**A2**, Associativity. .**A3**, Identity. There exists an identity (or neutral) element such that . The*identity*of is unique by Theorem 1.4 below.**A4**, Inverse. For each , there exists an inverse element such that . The*inverse*of is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:

**A5**, Commutativity. .

## Notation

The group is often referred to as "the group " or more simply as "" Nevertheless, the operation "" is fundamental to the description of the group. is usually read as "the group under ". When we wish to assert that is a group (for example, when stating a theorem), we say that " is a group under ".

The group operation can be interpreted in a great many ways. The generic notation for the group operation, identity element, and inverse of are respectively. Because the group operation associates, parentheses have only one necessary use in group theory: to set the scope of the inverse operation.

Group theory may also be notated:

**Additively**by replacing the generic notation by , with "+" being infix. Additive notation is typically used when numerical addition or a commutative operation other than multiplication interprets the group operation;**Multiplicatively**by replacing the generic notation by . Infix "*" is often replaced by simple concatenation, as in standard algebra. Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation.

Other notations are of course possible.

## Examples

### Arithmetic

### Function composition

- Let be an arbitrary set, and let be the set of all bijective functions from to . Let function composition, notated by infix , interpret the group operation. Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

## Alternative Axioms

The pair of axioms A3 and A4 may be replaced either by the pair:

**A3’**, left neutral. There exists an such that for all , .**A4’**, left inverse. For each , there exists an element such that .

or by the pair:

**A3”**, right neutral. There exists an such that for all , .**A4”**, right inverse. For each , there exists an element such that .

These evidently weaker axiom pairs are trivial consequences of A3 and A4. We will now show that the nontrivial converse is also true. Given a left neutral element and for any given then A4’ says there exists an such that .

*Proof*.
Let be an inverse of Then:

This establishes A4 (and hence A4”).

*Proof*.

This establishes A3 (and hence A3”).

**Theorem**: Given A1 and A2, A3’ and A4’ imply A3 and A4.

*Proof*. Theorems 1.2 and 1.2a.

**Theorem**: Given A1 and A2, A3” and A4” imply A3 and A4.

*Proof*. Similar to the above.

## Basic theorems

### Identity is unique

**Theorem 1.4**: The identity element of a group is unique.

*Proof*: Suppose that and are two identity elements of . Then

As a result, we can speak of *the* identity element of rather than *an* identity element. Where different groups are being discussed and compared, denotes the identity of the specific group .

### Inverses are unique

**Theorem 1.5**: The inverse of each element in is unique.

*Proof*: Suppose that and are two inverses of an element of . Then

As a result, we can speak of *the* inverse of an element , rather than *an* inverse. Without ambiguity, for all in , we denote by the unique inverse of .

### Inverting twice takes you back to where you started

**Theorem 1.6**: For all elements in a group .

*Proof*. and are both true by A4. Therefore both and are inverses of By Theorem 1.5,

Equivalently, inverting is an involution.

### Inverse of *ab*

**Theorem 1.7**: For all elements and in group , .

*Proof*. . The conclusion follows from Theorem 1.4.

### Cancellation

**Theorem 1.8**: For all elements in a group , then .

*Proof*.

(1) If , then multiplying by the same value on either side preserves equality.

(2) If then by (1)

(3) If we use the same method as in (2).

### Latin square property

**Theorem 1.3**: For all elements in a group , there exists a unique such that , namely .

*Proof*.

Existence: If we let , then .

Unicity: Suppose satisfies , then by Theorem 1.8, .

### Powers

**Theorem 1.9**: For all in group and :

## Order

### Of a group element

The order of an element *a* in a group *G* is the least positive integer *n* such that *a ^{n} = e*. Sometimes this is written "o(

*a*)=

*n*".

*n*can be infinite.

**Theorem 1.10**: A group whose nontrivial elements all have order 2 is abelian. In other words, if all elements *g* in a group *G* *g***g*=*e* is the case, then for all elements *a*,*b* in *G*, *a***b*=*b***a*.

*Proof*. Let *a*, *b* be any 2 elements in the group *G*. By A1, *a***b* is also a member of *G*. Using the given condition, we know that (*a***b*)*(*a***b*)=*e*. Hence:

*b***a*- =e*(
*b***a*)*e - = (
*a***a*)*(*b***a*)*(*b***b*) - =
*a**(*a***b*)*(*a***b*)**b* - =
*a***e***b* - =
*a***b*.

Since the group operation * commutes, the group is abelian.

### Of a group

The **order of the group G**, usually denoted by |

*G*| or occasionally by o(

*G*), is the number of elements in the set

*G*, in which case <

*G*,*> is a

*finite group*. If

*G*is an infinite set, then the group <

*G*,*> has order equal to the cardinality of

*G*, and is an

*infinite group*.

## Subgroups

A subset *H* of *G* is called a **subgroup** of a group <*G*,*> if *H* satisfies the axioms of a group, using the same operator "*", and restricted to the subset *H*. Thus if *H* is a subgroup of <*G*,*>, then <*H*,*> is also a group, and obeys the above theorems, restricted to *H*. The *order* of subgroup *H* is the number of elements in *H*.

A *proper subgroup* of a group *G* is a subgroup which is not identical to *G*. A *non-trivial* subgroup of *G* is (usually) any proper subgroup of *G* which contains an element other than *e*.

**Theorem 2.1**: If *H* is a subgroup of <*G*,*>, then the identity *e*_{H} in *H* is identical to the identity *e* in (*G*,*).

*Proof*. If *h* is in *H*, then *h***e*_{H} = *h*; since *h* must also be in *G*, *h***e* = *h*; so by theorem 1.8, *e*_{H} = *e*.

**Theorem 2.2**: If *H* is a subgroup of *G*, and *h* is an element of *H*, then the inverse of *h* in *H* is identical to the inverse of *h* in *G*.

*Proof*. Let *h* and *k* be elements of *H*, such that *h***k* = *e*; since *h* must also be in *G*, *h***h*^{ -1} = *e*; so by theorem 1.5, *k* = *h*^{ -1}.

Given a subset *S* of *G*, we often want to determine whether or not *S* is also a subgroup of *G*. A handy theorem valid for both infinite and finite groups is:

**Theorem 2.3**: If *S* is a non-empty subset of *G*, then *S* is a subgroup of *G* if and only if for all *a*,*b* in *S*, *a***b*^{ -1} is in *S*.

*Proof*. If for all *a*, *b* in *S*, *a***b*^{ -1} is in *S*, then

*e*is in*S*, since*a***a*^{ -1}=*e*is in*S*.- for all
*a*in*S*,*e***a*^{ -1}=*a*^{ -1}is in*S* - for all
*a*,*b*in*S*,*a***b*=*a**(*b*^{ -1})^{ -1}is in*S*

Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so *S* is subgroup.

Conversely, if *S* is a subgroup of *G*, then it obeys the axioms of a group.

- As noted above, the identity in
*S*is identical to the identity*e*in*G*. - By A4, for all
*b*in*S*,*b*^{ -1}is in*S* - By A1,
*a***b*^{ -1}is in*S*.

The intersection of two or more subgroups is again a subgroup.

**Theorem 2.4**: The intersection of any non-empty set of subgroups of a group *G* is a subgroup.

*Proof*. Let {*H*_{i}} be a set of subgroups of *G*, and let K = ∩{*H*_{i}}. *e* is a member of every *H*_{i} by theorem 2.1; so *K* is not empty. If *h* and *k* are elements of *K*, then for all *i*,

*h*and*k*are in*H*_{i}.- By the previous theorem,
*h***k*^{ -1}is in*H*_{i} - Therefore,
*h***k*^{ -1}is in ∩{*H*_{i}}.

Therefore for all *h*, *k* in *K*, *h***k*^{ -1} is in *K*. Then by the previous theorem, *K*=∩{*H*_{i}} is a subgroup of *G*; and in fact *K* is a subgroup of each *H*_{i}.

Given a group <*G*,*>, define *x***x* as *x*², *x***x***x**...**x* (*n* times) as *x*^{n}, and define *x*^{0} = *e*. Similarly, let *x*^{ -n} for (*x*^{ -1})^{n}. Then we have:

**Theorem 2.5**: Let *a* be an element of a group (*G*,*). Then the set {*a*^{n}: *n* is an integer} is a subgroup of *G*.

A subgroup of this type is called a *cyclic* subgroup; the subgroup of the powers of *a* is often written as <*a*>, and we say that *a* *generates* <*a*>.

## Cosets

If *S* and *T* are subsets of *G*, and *a* is an element of *G*, we write "*a***S*" to refer to the subset of *G* made up of all elements of the form *a***s*, where *s* is an element of *S*; similarly, we write "*S***a*" to indicate the set of elements of the form *s***a*. We write *S***T* for the subset of *G* made up of elements of the form *s***t*, where *s* is an element of *S* and *t* is an element of *T*.

If *H* is a subgroup of *G*, then a *left coset* of *H* is a set of the form *a***H*, for some *a* in *G*. A *right coset* is a subset of the form *H***a*.

If *H* is a subgroup of *G*, the following useful theorems, stated without proof, hold for all cosets:

- Any
*x*and*y*are elements of*G*, then either*x***H*=*y***H*, or*x***H*and*y***H*have empty intersection.

- Every left (right) coset of
*H*in*G*contains the same number of elements.

*G*is the disjoint union of the left (right) cosets of*H*.

- Then the number of distinct left cosets of
*H*equals the number of distinct right cosets of*H*.

Define the **index** of a subgroup *H* of a group *G* (written "[*G*:*H*]") to be the number of distinct left cosets of *H* in *G*.

From these theorems, we can deduce the important Lagrange's theorem, relating the order of a subgroup to the order of a group:

**Lagrange's theorem**: If*H*is a subgroup of*G*, then |*G*| = |*H*|*[*G*:*H*].

For finite groups, this can be restated as:

**Lagrange's theorem**: If*H*is a subgroup of a finite group*G*, then the order of*H*divides the order of*G*.

- If the order of group
*G*is a prime number,*G*is cyclic.

## See also

## References

- Jordan, C. R and D.A.
*Groups*. Newnes (Elsevier), ISBN 0-340-61045-X - Scott, W R.
*Group Theory*. Dover Publications, ISBN 0-486-65377-3