In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is defined as a non-empty set and a
binary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
called the
group operation.
As a shortcut is noted or even . This is called
infix notationInfix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation or postfix notation Infix notation is the common arithmetic and logical formula notation,...
.
The group must obey the following rules (or
axiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
s). Let be arbitrary elements of . Then:
- A1, Closure
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a member of the set...
. .
- A2, Associativity. .
- A3, Identity
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. There exists an identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
such that . The identity of is unique by Theorem 1.4 below.
- A4, Inverse
In abstract algebra, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is defined as a non-empty set and a
binary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
called the
group operation.
As a shortcut is noted or even . This is called
infix notationInfix notation is the common arithmetic and logical formula notation, in which operators are written infix-style between the operands they act on . It is not as simple to parse by computers as prefix notation or postfix notation Infix notation is the common arithmetic and logical formula notation,...
.
The group must obey the following rules (or
axiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
s). Let be arbitrary elements of . Then:
- A1, Closure
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a member of the set...
. .
- A2, Associativity. .
- A3, Identity
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. There exists an identity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
such that . The identity of is unique by Theorem 1.4 below.
- A4, Inverse
In abstract algebra, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
. For each , there exists an inverse elementIn abstract algebra, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
such that . The inverse of is unique by Theorem 1.5 below.
An
abelian groupAn abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
also obeys the additional rule:
Closure is part of the definition of "binary operation," so that
A1 is often omitted.
Notation
The group operation can be any of a number of operations. In generic terms, we denote the
group operation, identity element and inverse of : respectively.
There is also an
additive notation, where those are replaced by , and a
multiplicative notation where those are replaced by .
- The multiplicative notation is used for an actual multiplication, or a noncommutative operation.
- The additive notation is used for an actual addition, or any commutative operation except multiplication. The operator is always present in infix notation.
- Other notations might be used depending on the group.
The group is often referred to as "the group " or simply ""; but the operation "" is fundamental to the description of the group.
is usually pronounced "the group under ". When affirming that is a group (for example, in a theorem), we say that " is a group under ".
Arithmetic
- Take or or or , then is an abelian group.
- Take or or , then is an abelian group.
Function composition
- Take for the set of all bijective functions from to , where is an arbitrary set, then is a group. Its identity element is and the group inverse of a function is .
Alternative Axioms
A3 and A4 can be replaced by:
- A3’, left neutral. There exists an such that for all , .
- A4’, left inverse. For each , there exists an element such that .
Or alternatively by:
- A3”, right neutral. There exists an such that for all , .
- A4”, right inverse. For each , there exists an element such that .
These apparently weaker pairs of axioms are naturally implied by A3 and A4. We will now show that the converse is true.
Theorem: Assuming A1 and A2, A3’ and A4’ imply A3 and A4.
Proof. Let a left neutral element be given, and . By A4’ there exist an such that .
We proceed to show that .
Per A4’ there is an with:
Therefore:
This establishes A4 (and hence A4”).
This establishes A3 (and hence A3”).
Theorem: Assuming A1 and A2, A3” and A4” imply A3 and A4.
Proof. Similar to the above.
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 .
Latin square property
Theorem 1.3: For all elements
a,
b in
G, there exists a unique
x in
G such that
a*
x =
b.
Proof. At least one such
x surely exists, for if we let
x =
a -1*
b, then
x is in
G (by A1, closure) and:
- 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.
Inverting twice gets you back where you started
Theorem 1.6: For all elements
a in group
G, (
a -1)
-1=
a.
Proof.
a -1*
a =
e. The conclusion follows from Theorem 1.4.
Inverse of ab
Theorem 1.7: For all elements
a,
b in group
G, (
a*
b)
-1=
b -1*
a -1.
Proof. (
a*
b)*(
b -1*
a -1) =
a*(
b*
b -1)*
a -1 =
a*e*
a -1 =
a*
a -1 =
e. The conclusion follows from Theorem 1.4.
Cancellation
Theorem 1.8: For all elements
a,
x, and
y in group
G, if
a*
x=
a*
y, then
x=
y; and if
x*
a=
y*
a, then
x=
y.
Proof. 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
Of a group element
The order of an element
a in a group
G is the least positive integer
n such that
an = 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
abelianAn abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
. 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
abelianAn abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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 cardinalityIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
of G, and is an infinite group.
Subgroups
A subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
H of G is called a subgroupIn the mathematical subject known as group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
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 cyclicIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
subgroup; the subgroup of the powers of a
is often written as <a
>, and we say that a
generates
In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses....
<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
cosetIn mathematics, if G is a group, H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
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:
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 indexIn mathematics, specifically group theory, the index of a subgroup H in a group G is the “relative size” of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively “half” of the elements of G lie in H...
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
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
: If
H
is a subgroup of G
, then |G
| = |H
|*[G
:H
].
For finite groups, this can be restated as:
H
is a subgroup of a finite group G
, then the order of H
divides the order of G
.
G
is a prime number, G is cyclic.
See also
- group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
- abelian groups
- Glossary of group theory
A group is a set G closed under a binary operation • satisfying the following 3 axioms:* Associativity: For all a, b and c in G, • c = a • ....
- List of group theory topics