Table of contents  
Introduction  Groups  
Cyclic groups  
Permutation groups  
Other examples 
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Definition  A group (G,·) is a nonempty set G together with a binary operation · on G such that the following conditions hold:
(i) Closure: For all a,b G the element a · b is a uniquely defined element of G.
(ii) Associativity: For all a,b,c ∈ G, we have
a · (b · c) = (a · b) · c.
(iii) Identity: There exists an identity element e ∈ G such that:
e · a = a and a · e = a, for all a ∈ G.
(iv) Inverses: For each a G there exists an inverse element a^{1} ∈ G such that:
a · a^{1} = e and a^{1} · a = e.
We will usually simply write ab for the product a · b.
Proposition 1 (Cancellation Property for Groups): Let G be a group, and let a,b,c ∈ G.
(a) If ab=ac, then b=c.
(b) If ac=bc, then a=b.
Definitions
Proposition 2: Let G be a group with identity element e, and let H be a subset of G. Then H is a subgroup of G if and only if the following conditions hold:
Theorem (Lagrange): If H is a subgroup of the finite group G, then the order of H is a divisor of the order of G.
Corollary 1: Let G be a finite group of order n.
Example (Euler's theorem): Let G be the multiplicative group of congruence classes modulo n. The order of G is given by (n), and so by Corollary 1, raising any congruence class to the power (n) must give the identity element.
Corollary 2: Any group of prime order is cyclic.
Isomorphism: Let G_{1} and G_{2} be groups, and let θ: G_{1} > G_{2} be a function. Then is said to be a group isomorphism if
In this case, G_{1} is said to be isomorphic to G_{2}, and this is denoted by G_{1} ≌ G_{2}.
Proposition 3: Let θ: G_{1} > G_{2} be an isomorphism of groups.
Let G be a group, and let a be any element of G. The set is called the cyclic subgroup generated by a. <a> = {x ∈ G  x = a^{n} for some n ∈ Z}
Group G is called a cyclic group if there exists an element a G such that G=<a>. In this case, a is called a generator of G.
Proposition 4: Let G be a group, and let a ∈ G.
Proposition 5: Let a be an element of group G.
Corollaries to Lagrange's Theorem (restated):
(a) For any a ∈ G, o(a) is a divisor of G.
(b) For any a ∈ G, a^{n} = e, for n = G.
(c) Any group of prime order is cyclic.
Theorem: Every subgroup of a cyclic group is cyclic.
Theorem: Let G cyclic group.
Proposition 6: Let G = <a> be a cyclic group with G = n.
(a) If m Z, then <a^{m}> = <a^{d}>, where d=gcd(m,n), and am has order n/d.
(b) The element a^{k} generates G if and only if gcd(k,n)=1.
(c) The subgroups of G are in onetoone correspondence with the positive divisors of n.
(d) If m and k are divisors of n, then <a^{m}> ⊂ <a^{k}> if and only if k  m.
Exponent of G: Let G be a group. If there exists a positive integer N such that a^{N}= e for all a ∈ G, then the smallest such positive integer is called the exponent of G.
Lemma: Let G be a group, and let a,b ∈ G be elements such that ab = ba. If the orders of a and b are relatively prime, then o(ab) = o(a)o(b).
Proposition 7: Let G be a finite abelian group.
(a) The exponent of G is equal to the order of any element of G of maximal order.
(b) Group G is cyclic if and only if its exponent is equal to its order.
A permutation group is a finite group G whose elements are permutations of a given set and whose group operation is composition of permutations in G. Permutation groups have orders dividing n!.
The set of all permutations of a set S is denoted by Sym(S). The set of all permutations of the set {1,2,...,n} is denoted by S_{n}.
Proposition 8: If S is any nonempty set, then Sym(S) is a group under the operation of the composition of functions.
Theorem: Every permutation in S_{n} can be written as a product of disjoint cycles. The cycles that appear in the product are unique.
Proposition 9: If a permutation in S_{n} is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.
Theorem: (Cayley) Every group is isomorphic to a permutation group.
Dihedral Group: Let n > 2 be an integer. The group of rigid motions of a regular ngon is called the nth dihedral group, denoted by Dn.
We can describe the nth dihedral group as
D_{n} = {a^{k}, a^{k}b  0 < k < n},
subject to the relations o(a) = n, o(b) = 2, and ba = a^{1}b.
Theorem: If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases.
Even and Odd Permutation: A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions.
Proposition 10: The set of all even permutations of S_{n} is a subgroup of S_{n}.
Alternating Group: The set of all even permutations of S_{n} is called the alternating group on n elements and will be denoted by A_{n}.
(Group of units modulo n) Let n be a positive integer. The set of units modulo n, denoted by Z_{n}^{×}, is an abelian group under the multiplication of congruence classes. Its order is given by the value φ(n) of Euler's phifunction.
General Linear Group. The set of all invertible n × n matrices with entries in R is called the general linear group of degree n over the real numbers and is denoted by GL_{n}(R).
Proposition. The set GL_{n}(R) forms a group under matrix multiplication.
Definition. Let G_{1} and G_{2} be groups. The set of all ordered pairs (x_{1},x_{2}) such that x_{1} ∈ G_{1} and x_{2} ∈ G_{2} is called the direct product of G_{1} and G_{2}, denoted by G_{1} × G_{2}.
Proposition. Let G_{1} and G_{2} be groups.
Field: Let F be a set with two binary operations + and · with respective identity elements 0 and 1, where 1 is distinct from 0. Then F is called a field if
(i) the set of all elements of F is an abelian group under +;
(ii) the set of all nonzero elements of F is an abelian group under ·;
(iii) a · (b+c) = a · b + a · c for all a,b,c in F.
General Linear Group of Degree n: Let F be a field. The set of all invertible n × n matrices with entries in F is called the general linear group of degree n over F, and is denoted by GL_{n}(F).
Proposition. Let F be a field. Then GL_{n}(F) is a group under matrix multiplication.
Proposition. If m,n are positive integers such that gcd(m,n)=1, then
Example. 3.3.7. (Quaternion group)
Consider the following set of invertible 2 × 2 matrices with entries in the field of complex numbers.
If we let
then we have the identities
i^{2} = j^{2} = k^{2} = 1;
ij = k, jk = i, ki = j;
ji = k, kj = i, ik = j.
These elements form a nonabelian group Q of order 8 called the quaternion group, or group of quaternion units.
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