# Elementary group theory

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In mathematics, a group (*G*,*) is usually defined as:

*G* is a set and * is an associative binary operation on *G*, obeying the following rules (or axioms):

- A1. ( Closure) If
*a*and*b*are in*G*, then*a***b*is in*G* - A2. ( Associativity) If
*a*,*b*, and*c*are in*G*, then (*a***b*)**c*=*a**(*b***c*). - A3. ( Identity)
*G*contains an element, often denoted*e*, such that for all*a*in*G*,*a***e*=*e***a*=*a*. We call this element the identity of (*G*,*). (We will show*e*is unique later.) - A4. ( Inverses) If
*a*is in*G*, then there exists an element*b*in*G*such that*a***b*=*b***a*=*e*. We call*b*the inverse of*a*. (We will show*b*is unique later.)

Closure and associativity are part of the definition of "associative binary operation", and are sometimes omitted, particularly closure.

Notes:

- The * is not necessarily multiplication. Addition works just as well, as do many less standard operations.
- When * is a standard operation, we use the standard symbol instead (for example, + for addition).
- When * is addition or any commutative operation (except multiplication), the identity is usually denoted by 0 and the inverse of
*a*by -*a*. The operation is always denoted by something other than *, often +, to avoid confusion with multiplication. - When * is multiplication or any non-commutative operation, the identity is usually denoted by 1 and the inverse of
*a*by*a*^{ -1}. The operation is often omitted,*a***b*is often written*ab*. - (
*G*,*) is usually pronounced "the group*G*under *". When affirming that it is a group (for example, in a theorem), we say that "*G*is a group under *". - The group (
*G*,*) is often referred to as "the group*G*" or simply "*G*"; but the operation "*" is fundamental to the description of the group.

## Examples

### (*R*,+) is a group

The real numbers (*R*) are a group under addition (+).

- Closure: Clear; adding any two numbers gives another number.
- Associativity: Clear; for any
*a*,*b*,*c*in*R*, (*a*+*b*)+*c*=*a*+(*b*+*c*). - Identity: 0. For any
*a*in*R*,*a*+0=*a*. (Hence the denotation 0 for identity) - Inverses: For any
*a*in*R*, -*a*+*a*=0. (Hence the denotation -*a*for inverse)

### (*R*,*) is not a group

The real numbers (*R*) are NOT a group under multiplication (*).

- Identity: 1.
- Inverses: 0*
*a*=0 for all*a*in*R*, so 0 has no inverse.

### (*R*^{#},*) is a group

^{#}

The real numbers without 0 (*R ^{#}*) are a group under multiplication (*).

- Closure: Clear; multiplying any two numbers gives another number.
- Associativity: Clear; for any
*a*,*b*,*c*in*R*, (*a***b*)**c*=*a**(*b***c*). - Identity: 1. For any
*a*in*R*,*a**1=*a*. (Hence the denotation 1 for identity) - Inverses: For any
*a*in*R*,*a*^{ -1}**a*=1. (Hence the denotation*a*^{ -1}for inverse)

## Basic theorems

### Inverse relations are commutative

**Theorem 1.1**: For all *a* in *G*, *a*^{ -1}**a* = *e*.

- By expanding
*a*^{ -1}**a*, we get*a*^{ -1}**a*=*a*^{ -1}**a***e*(by A3')*a*^{ -1}**a***e*=*a*^{ -1}**a**(*a*^{ -1}*(*a*^{ -1})^{ -1}) (by A4',*a*^{ -1}has an inverse denoted (*a*^{ -1})^{ -1})*a*^{ -1}**a**(*a*^{ -1}*(*a*^{ -1})^{ -1}) =*a*^{ -1}*(*a***a*^{ -1})*(*a*^{ -1})^{ -1}=*a*^{ -1}*e*(*a*^{ -1})^{ -1}(by associativity and A4')*a*^{ -1}**e**(*a*^{ -1})^{ -1}=*a*^{ -1}*(*a*^{ -1})^{ -1}=*e*(by A3' and A4')

- Therefore,
*a*^{ -1}**a*=*e*

### Identity relations are commutative

**Theorem 1.2**: For all *a* in *G*, *e***a* = *a*.

- Expanding
*e***a*,*e***a*= (*a***a*^{ -1})**a*(by A4)- (
*a***a*^{ -1})**a*=*a**(*a*^{ -1}**a*) =*a***e*(by associativity and the previous theorem) *a***e*=*a*(by A3)

- Therefore
*e***a*=*a*

### Latin square property

**Theorem 1.3**: For all *a*,*b* in *G*, there exists a unique *x* in *G* such that *a***x* = *b*.

- Certainly, at least one such
*x*exists, for if we let*x*=*a*^{ -1}**b*, then*x*is in*G*(by A1, closure); and then*a***x*=*a**(*a*^{ -1}**b*) (substituting for*x*)*a**(*a*^{ -1}**b*) = (*a***a*^{ -1})**b*(associativity A2).- (
*a***a*^{ -1})**b*=*e***b*=*b*. (identity A3). - Thus an
*x*always exists satisfying*a***x*=*b*.

- To show that this is unique, if
*a***x*=*b*, then*x*=*e***x**e***x*= (*a*^{ -1}**a*)*x- (
*a*^{ -1}**a*)*x =*a*^{ -1}*(*a***x*) *a*^{ -1}*(*a***x*) =*a*^{ -1}**b*- Thus,
*x*=*a*^{ -1}**b*

Similarly, for all *a*,*b* in *G*, there exists a unique *y* in *G* such that *y***a* = *b*.

### The identity is unique

**Theorem 1.4**: The identity element of a group (*G*,*) is unique.

*a***e*=*a*(by A3)- Apply theorem 1.3, with
*b*=*a*.

Alternative proof: Suppose that *G* has two identity elements, *e* and *f* say. Then *e***f* = *e*, by A3', but also *e***f* = *f*, by Theorem 1.2. Hence *e* = *f*.

As a result, we can speak of *the* identity element of (*G*,*) rather than *an* identity element. Where different groups are being discussed and compared, often *e*_{G} will be used to identify the identity in (*G*,*).

### Inverses are unique

**Theorem 1.5**: The inverse of each element in (*G*,*) is unique; equivalently, for all *a* in *G*, *a***x* = *e* if and only if *x*=*a*^{ -1}.

- If
*x*=*a*^{ -1}, then*a***x*=*e*by A4. - Apply theorem 1.3, with
*b*=*e*.

Alternative proof: Suppose that an element *g* of *G* has two inverses, *h* and *k* say. Then *h* = *h***e* = *h**(*g***k*) = (*h***g*)**k* = *e***k* = *k* (equalities justified by A3'; A4'; A2; Theorem 1.1; and Theorem 1.2, respectively).

As a result, we can speak of *the* inverse of an element *x*, rather than *an* inverse.

### Inverting twice gets you back where you started

**Theorem 1.6**: For all *a* belonging to a group (*G*,*), (*a*^{ -1})^{ -1}=*a*.

*a*^{ -1}**a*=*e*.- Therefore the conclusion follows from theorem 1.4.

### The inverse of ab

**Theorem 1.7**: For all *a*,*b* belonging to a group (*G*,*), (*a***b*)^{ -1}=*b*^{ -1}**a*^{ -1}.

- (
*a***b*)*(*b*^{ -1}**a*^{ -1}) =*a**(*b***b*^{ -1})**a*^{ -1}=*a**e**a*^{ -1}=*a***a*^{ -1}= e - Therefore the conclusion follows from theorem 1.4.

### Cancellation

**Theorem 1.8**: For all *a*,*x*,*y*, belonging to a group (*G*,*), if *a***x*=*a***y*, then *x*=*y*; and if *x***a*=*y***a*, then *x*=*y*.

- If
*a***x*=*a***y*then:*a*^{ -1}*(*a***x*) =*a*^{ -1}*(*a***y*)- (
*a*^{ -1}**a*)**x*= (*a*^{ -1}**a*)**y* *e***x*=*e***y**x*=*y*

- If
*x***a*=*y***a*then- (
*x***a*)**a*^{ -1}= (*y***a*)**a*^{ -1} *x**(*a***a*^{ -1}) =*y**(*a***a*^{ -1})*x***e*=*y***e**x*=*y*

- (

### Repeated use of *

**Theorem 1.9**: For every *a* in a group (G,*), we define

as :

and

and

However, when the operation is noted *+*, we note

as :

and

Where (This generalizes the associativity.)

### Groups in which all non-trivial elements have order 2

**Theorem 1.10**: A group where all non-trivial elements have order 2 is abelian. In other words, if all elements *g* in a group G satisfy *g***g*=*e*, then for any 2 elements *g*, *h* in G, *g***h*=*h***g*.

- Let
*g*,*h*be any 2 elements in a group G - By A1,
*g***h*is also a member of G - Using the given condition, we know (
*g***h*)*(*g***h*)=*e*. Now*g**(*g***h*)*(*g***h*) =*g***e**g**(*g***h*)*(*g***h*)**h*=*g***e***h*- (
*g***g*)*(*h***g*)*(*h***h*) = (*g***e*)**h* *e**(*h***g*)**e*=*g***h**h***g*=*g***h*

- Since the group operation commutes, the group is abelian

## Definitions

Given a group (*G*, *), if the total number of elements in *G* is finite, then the group is called a *finite group*. The **order of a group** (*G*,*) is the number of elements in *G* (for a finite group), or the cardinality of the group if *G* is not finite. The order of a group *G* is written as |*G*| or (less frequently) o(*G*).

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*,*).

- If
*h*is in*H*, then*h***e*_{H}=*h*; since*h*must also be in*G*,*h***e*=*h*; so by theorem 1.4,*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*.

- 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*. One handy theorem that covers the case for both finite and infinite 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*.

- 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*.

- As noted above, the identity in

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.

- 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}.

In a group (*G*,*), define *x*^{0} = *e*. We write *x***x* as *x*² ; and in general, *x***x***x**...**x* (*n* times) as *x*^{n}. Similarly, we write *x*^{ -n} for (*x*^{ -1})^{n}.

**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*>.

If there exists a positive integer *n* such that *a*^{n}=*e*, then we say the element *a* has **order** *n* in G where *n* is the smallest *n*. Sometimes this is written as "o(*a*)=*n*".

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*.

Some useful theorems about cosets, stated without proof:

**Theorem**: If *H* is a subgroup of *G*, and *x* and *y* are elements of *G*, then either *x***H* = *y***H*, or *x***H* and *y***H* have empty intersection.

**Theorem**: If *H* is a subgroup of *G*, every left (right) coset of *H* in *G* contains the same number of elements.

**Theorem**: If *H* is a subgroup of *G*, then *G* is the disjoint union of the left (right) cosets of *H*.

**Theorem**: If *H* is a subgroup of *G*, then the number of distinct left cosets of *H* is the same as 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 also allows us to state:

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

**Theorem**: If the order of a group *G* is a prime number, then the group is cyclic.