In mathematics, a group (G,*) is usually defined as: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

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=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=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. For closure in computer science, see closure (computer science). ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...

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.

Mathematical meaning In mathematics, especially abstract algebra, a binary operation on a set S is commutative if for all x and y in S. Otherwise, the operation is noncommutative. ...

Examples

(R,+) is a group

The real numbers (R) are a group under addition (+). Please refer to Real vs. ...

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 (*). Please refer to Real vs. ...

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 (*). Please refer to Real vs. ...

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

Inverses work on either side

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

An identity works on either side

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. In mathematics, the Latin square property is an elementary property of all groups. ...

• 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, a^maⁿ = a^m+n = aⁿa^m and (a^m)ⁿ = (aⁿ)^m = a^nm. (This generalizes the associativity.)

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). In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

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 is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In 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 is a group...

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.

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 x⁰ = e. We write x*x as x² ; and in general, x*x*x*...*x (n times) as xⁿ. Similarly, we write x ^-n for (x ^-1)ⁿ.

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 cyclic subgroup; the subgroup of the powers of a is often written as <a>, and we say that a generates <a>. In 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 a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... 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. ...

If there is a positive integer n such that aⁿ=e, then we say the element a has order n in G. 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. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

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: In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H...

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.

References

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