Elementary group theory
Encyclopedia
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 axiom
s). Let
be arbitrary elements of 
. Then:
An abelian group
also obeys the additional rule:
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:
Other notations are of course possible.
or by the pair:
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 
.
Theorem 1.2:
Proof.
Let
be an inverse of 
Then:
This establishes A4 (and hence A4”).
Theorem 1.2a:
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.
is unique.
Proof: Suppose that
and 
are two identity elements of 
. Then
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a group
Group (mathematics)
In 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 the algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
, where is a non-empty set and denotes a binary operationBinary operation
In 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. The notation is normally shortened to the infix notationInfix notation
Infix 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,...
, or even to .A group must obey the following rules (or axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s). Let
be arbitrary elements of . Then:
- A1, ClosureClosure (mathematics)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 unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
.
. This axiom is often omitted because a binary operation is closed by definition. - A2, Associativity.
. - A3, IdentityIdentity 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...
. There exists an identity (or neutral) element
such that 
. The identity of 
is unique by Theorem 1.4 below. - A4, InverseInverse elementIn abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a 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 elementInverse elementIn abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a 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 group
Abelian group
In abstract algebra, an 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:
- 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 thegroup 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 infixInfixAn infix is an affix inserted inside a word stem . It contrasts with adfix, a rare term for an affix attached to the end of a stem, such as a prefix or suffix.-Indonesian:...
. 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.
Arithmetic
- Take
or 
or 
or 
, then 
is an abelian group. - Take
or 
or 
, then 
is an abelian group.
Function composition
- Let
be an arbitrary set, and let 
be the set of all bijective functions from 
to 
. Let function compositionFunction compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
, notated by infixInfixAn infix is an affix inserted inside a word stem . It contrasts with adfix, a rare term for an affix attached to the end of a stem, such as a prefix or suffix.-Indonesian:...
, 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 .Theorem 1.2:
Proof.
Let
be an inverse of Then:This establishes A4 (and hence A4”).
Theorem 1.2a:
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.
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
, 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
For
and 
in group 
we define:
-
Theorem 1.9: For all
in group 
and 
:
-
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 abelianAbelian groupIn abstract algebra, an 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:- =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 abelianAbelian groupIn abstract algebra, an 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 cardinality of G, and is an infinite group.
Subgroups
A subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
H of G is called a subgroupSubgroupIn group theory, given a group G under a binary operation *, a subset H of G is called 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 eH in H is identical to the identity e in (G,*).
Proof. If h is in H, then h*eH = h; since h must also be in G, h*e = h; so by theorem 1.8, eH = 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 {Hi} be a set of subgroups of G, and let K = ∩{Hi}. e is a member of every Hi 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 Hi.
- By the previous theorem, h*k -1 is in Hi
- Therefore, h*k -1 is in ∩{Hi}.
Therefore for all h, k in K, h*k -1 is in K. Then by the previous theorem, K=∩{Hi} is a subgroup of G; and in fact K is a subgroup of each Hi.
Given a group <G,*>, define x*x as x², x*x*x*...*x (n times) as xn, and define x0 = 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 {an: n is an integer} is a subgroup of G.
A subgroup of this type is called a cyclicCyclic groupIn 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 generatesGenerating set of a groupIn abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
<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 cosetCosetIn mathematics, if G is a group, and 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:
- And 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 indexIndex of a subgroupIn 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 theoremLagrange's theorem (group theory)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:
- 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
- group theoryGroup theoryIn 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 theoryGlossary of group theoryA 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
-
-