Groups

Seite 1

1. Definitions and Examples

Definition 1.1 - A group (G, .) is a set G on which an operation, often called the product or multiplication, is defined (i.e. for all a,b in G, the product of a and b, a.b, exists and is an element of G) for which the following Group Axioms are satisfied.

(G1) Associativity Axiom - For all a, b, c in G, (a.b).c = a.(b.c).

(G2) Identity Axiom - There exists an element 1 in G such that for all a in G, 1.a = a.1 = a, i.e. G has an identity element.

(G3) Inverses Axiom - For every a in G there exists and element a' in G such that a.a' = 1 = a'.a, i.e. each element is an inverse.

Addition can be used instead of multiplication as the group operation.

Definition 1.4 - If two elements a, b in a group G have the property that ab = ba then a and b are said to commute. If all pairs of elements in G commute i.e. ab = ba for all a, b in G, then G is called commutative or abelian. A group which is not abelian is called non-commutative or non-abelian.

Definition 1.6 - An infinite group is a group with infinitely many elements. A finite group is a group with only finitely many elements, for such a group G, the number of elements |G| is called the order of G.

2. Some Basic Properties of Groups

Proposition 1.9 (Uniqueness of Identity) - The group G contains only one identity element.

Proposition 1.10 (Uniqueness of Inverses) - In the group G, each element has only one inverse.

Proposition 1.11 (Rules for Inverses) - Let x, y in G. Then

1) 1^-1 = 1

2) (x^-1)^-1 = x

3) (xy)^-1 = y^-1x^-1 (the reversal rule).

Proposition 1.12 (Solutions of Equations) - Let a, b, x, y be in G. Then

1) ax = b if and only if x = a^-1 b

2) ya = b if and only if y = ba^-1.

Proposition 1.13 (Cancellation Laws) - Let u, v, x, y be in G.

1) If ux = uy then x = y.

2) If xv = yv then x = y.

Proposition 1.14 (One-sided Test for Inverse) - Let a, b, c be in G.

1) If ab = 1 then b = a^-1.

2) If ca = 1 then c = a^-1.

Definition 1.16 (Powers of an Element) - Let x be in G and let n be an integer. We define the n-th power of x, x^n, as follows:

x^n = xx...x (n factors) if n>0, 1 if n=0, (x^-1)^-n if n

Proposition 1.17 (Index Laws) - Let x be in G and m, n in Z. Then x^mx^n = x^n+m, (x^m)^n = x^mn. Be careful with these since (xy)^2 may not equal x^2y^2.

Definition 1.19 - Let a, x be in G. Then the element axa^-1 is called the conjugate of x by a. Any element y in G that can be written as y = axa^-1 for some a in G is called a conjugate of x and we then say that x and y are conjugate.

3. Subgroups

Definition 1.21 - If H is itself a group with the same product as G, then H is called a subgroup of G, denoted by H

Definition 1.23 - The subgroup {1}

Lemma 1.24 (Identity and Inverses in a Subgroup) - Let H

1) the identity 1_H of H is the same as the identity 1_G of G i.e. 1_H = 1_G.

2) for each h in H the inverse of h in the group H is the same as the inverse of h in the group G.

Proposition 1.25 (Test for a Subgroup) - Let H be a subset of the group G. Then H is a subgroup if and only if all of the following hold.

1) 1_G is an element of H.

2) H is closed under multiplication i.e. if x, y are in H then xy is in H.

3) H is closed under taking inverses i.e. if x is in H then x_1 is in H.

Proposition 1.27 (Alternative Test for a Subgroup) - Let H be a subset of the group G. Then H is a subgroup if and only if the following hold.

1) H is not the empty set.

2) If x, y are in H, then x^-1y is in H.

Proposition 1.28 (Subgroup Test for Abelian Group) - Let H be a subset of the group G. Then H is a subgroup if and only if both of the following hold;

1) H is not the empty set.

2) H is closed under subtraction i.e. if x, y are in H then x-y is in H.

Proposition 1.29 - Let H

Proposition 1.31 - The intersection of any collection of subgroups of H_q

Definition 1.32 - For a in G, the centraliser of a in G, C_G(a) is the set of all elements in G that commute with a.

Proposition 1.33 - For each a in G, C_G(a) is a subgroup of G, C_G(a)

Definition 1.36 - The centre of G is the set of all elements of G that commute with every element of G.

Proposition 1.37 - For any group G, the centre of G, Z(G), is a subset of G.

Lemma 1.39 - For any group G, the subgroup Z(G)

Lemma 1.40 - G is abelian if and only if G=Z(G).

4. Cyclic Subgroups and Cyclic Groups

Throughout, let G be a group.

Definition 1.41 - For a in G, let = {a^m: m in Z} c=G.

Proposition 1.42 - For a in G,

Definition 1.43 - The group G is called cyclic if it is a cyclic subgroup of itself, i.e. there is an element g in G such that G=. If G=, then g is called a generator of G.

Proposition 1.45 - Let a be in G and let H is equal to the intersection of all the subgroups of G which contain a.

5. The Order of an Element

Definition 1.46 - Let a be in G

1) If no positive power of a equals 1 then we say that a has infinite order or that the order of a is infinite.

2) If a^k = 1 for some positive integer k we say that a has finite order; in this care if k is the smallest such positive integer we say that a has order k or that k is the order of a, and we write o(a)=k or |a|=k. In particular, the identity is the only element of order 1 because the equation a^1=1 holds if and only if a=1.

Proposition 1.50 - Let a in G have infinite order. Then

1) for m, n in Z, if a^m = a^n then m=n;

2) the cyclic subgroup is infinite.

Corollary 1.51 - In a finite group every element has finite order.

Proposition 1.52 - Let a be in G with o(a)=k. Then the following hold.

1) If m in Z has remainder r on division by k then a^m = a^r.

2) The subgroup has order K and 1, a, a^2, ..., a^k-1 is a complete repetition-free list of its elements.

3) For m in Z, a^m = 1 if and only if k divides m.

Corollary 1.53 - In any group the order of an element a is the same as the order of the subgroup ie o(a) = ||.

Corollary 1.55 - In a finite group G, the order of each element is less than or equal to |G|.

Proposition 1.57 - Supposed that G is a finite group of order n and let x be in G. Then G is cyclic with G = if and only if o(x) = n. If g is a generator of the cyclic group G = of order n, then g^r is also a generator if and only if gcd(r, n)=1.

Proposition 1.59 - Every cyclic group is abelian.

Theorem 1.60 - Every subgroup of a cyclic group is cyclic.

6. Cosets

Definition 1.62 - Let X, Y be subsets of G. Then XY={xy : x in X, y in Y}, that is the set of all elements of G that can be expressed in the form (an element of X).(an element of Y). The order is important here, also XY = the empty set if X is the empty set or Y is the empty set.

Definition 1.63 - Let H

Proposition 1.66 - With the above notation, if H is finite then |xH|=|H|= |Hx|.

Definition 1.67 - Left congruence modulo and Right congruence modulo.

Proposition 1.69 - Left and right congruence modulo H are equivalence relations on G.

Proposition 1.70 - For x in G, the left congruence class of x modulo H is the coset xH.

Corollary 1.71 - The left cosets of H partition G into disjoint subsets of equal cardinality |H|.

Proposition 1.72 - Let x, y be in G. Then xH = yH implies y^-1x is in H.

Corollary 1.73 - Let x be in G. Then xH=H implies x is in H.

Definition 1.75 - Let H be a subgroup of a group G. The cardinality of the set G/H of all left cosets of H in G is called the (left) index of H in G, denote |G:H|.

Theorem 1.76 (Lagrange's Theorem) - Let G be a finite group and left H

Proposition 1.79 - Let H and K be subgroups of a finite group G such that K

Theorem 1.80 - Let G be a finite group. Then for any element g in G, o(g) divides |G|, and g^|G|=1.

Theorem 1.81 - Every group of prime order is cyclic. Furthermore, in a group of prime order p, every non-identity element has order p and is a generator of the group.

Theorem 1.82 - Let G be a finite group and let p be any prime. Then the number of elements of G of order p is m(p-1) where m is the number of subgroups of G of order p.

7. Normal Subgroups

Definition 1.83 - For a subgroup H

Proposition 1.84 - Let H

Definition 1.85 - Let H

Proposition 1.86 - Let H

Proposition 1.87 - Let H

Proposition 1.89 - Let G be a group.

a) The subgroups {1}

b) The centre Z(G)

c) If G is abelian, then every subgroup H

Proposition 1.90 - Let G be a group. If there is a collection of normal subgroups in G, then the intersection of these subgroups is also a normal subgroup.

Proposition 1.91 - Let H

Definition 1.92 - Two subgroups H

Proposition 1.93 - Let H and K be subgroups of G. Then HK is a subgroup of G if and only if H and K are permutable.

Proposition 1.94 - Let H and K be subgroups of G and supposed that K is normal to G. The KH=HK and so HK

8. Quotient Groups

Definition 1.96 - Given a subgroup H

Proposition 1.97 - Supposed K is a normal subgroup of G. Then the rule (xK)(yK)=(xy)K for all x and y in G provides a well-defined multiplication on G/K, G/K x G/K -> G/K; (xK)(yK)=(xy)K.

Theorem 1.98 - Supposed that K is a normal subgroup of G. Then the set of left cosets of K in G, G/K, forms a group under the multiplication defined in Proposition 1.97.

Definition 1.99 - The group G/K defined in Theorem 1.98 is called the quotient group of G by K. Sometimes the phrase factor group is used instead of quotient groups.

Theorem 1.103 - For any group G, the factor group G/Z(G) cannot be a non-trivial cyclic group.

Theorem 1.104 - If p is a prime then every p-group has a non-trivial centre.

Corollary 1.105 - If p is prime, then every group of order p^2 is abelian.

Theorem 1.106 (Correspondence Theorem) - Let G be a group and let K be a normal subgroup of G

1) If H is a subgroup of G containing K, then K is a normal subgroup of H and H/K is a subgroup of G/K.

2) If H and H' are subgroups of G which contain K and such that H/K = H'/K, then H = H'.

3) If H is a subgroup of G which contains K, then H is a normal subgroup of G if and only if H/K is a normal subgroup of G/K.

4) Every subgroup of G/K has the form H/K for some subgroup H of G satisfying K

9. Homomorphisms

Definition 1.108 - Let G and G' be groups. A function t:G->G' is called a (group) homomorphism, if for all x, y in G, t(xy)= t(x)t(y)

Proposition 1.112 - Let t:G->G' be a homomorphism between two groups with identity elements 1=1_G and 1'=1_G'.

a) t preserves identity elements ie t(1)=1'.

b) t preserves inverses ie for all g in G, t(g^-1)= t(g)^-1

c) For g_1, g_2, ...., g_n in in G, t(g_1g_2...g_n) = t(g_1)t(g_2)...t(g_n)

d) For g in G and m in Z, t(g^m) = t(g)^m.

Proposition 1.113 - Let t:G->G' be a homomorphism between two groups with identity elements as above.

a) If H

b) If H'

Corollary 1.114 - The imagine imt=tG is a subgroup of G' ie imt

Definition 1.115 - The kernel of the homomorphism t:G->G' is kert = t^-1{1'} = {g in G: t(g) = 1'} which is a subset of G.

Proposition 1.116 - The kernel of t:G->G' is a normal subgroup of G.

Definition 1.118 - A surjective homomorphism is called an epimorphism.

- An injective homomorphism is called a monomorphism.

- A bijective homomorphism is called an isomorphism.

- A homomorphism t:G->G is called an endomorphism of G.

- An isomorphism t:G->G is called an automorphism of G. The set of all automorphisms of G is denoted Aut(G).

Proposition 1.119 - Let K be a normal subgroup of G. The quotient function p:G->G/K is an epimorphism and ker(p)=K.

Proposition 1.121 (Test for injectivity of a homomorphism) - Let t:G->G' be a homomorphism. Then t is a monomorphism if and only if ker(t)={1}.

Propositiove 1.122 - Let G, G' and G'' be groups and let t:G->G' and u:G'->G'' be homomorphisms. Then u o t :G->G'' is a homomorphism.

Proposition 1.123 - Let G and G' be groups and let t:G->G' be an isomorphism. Then the inverse function t^-1:G'->G is also an isomorphism.

10. Isomorphisms and the Isomorphism Theorems

Definition 1.124 - Let G and G' be groups. If there exists an isomorphism t:G->G', then G is said to be isomorphic to G', indicated by writing G~=G'.

Proposition 1.125 - The relation of being isomorphic ~= has the properties of an equivalence relation on any set of groups.

Proposition 1.126 - Let G and G' be groups and supposed that t:G->G' is a monomorphism ie an injective homomorphism. Then im(t)~=G.

Theorem 1.127 (Cayley's Theorem) - Every group is isomorphic to a group of permutations ie for every group G there is a set X and a subgroup H

Theorem 1.128 (First Isomorphism Theorem) - Let G and G' be groups and supposed that t:G->G' is a homomorphism. Then G/ker(t) ~= im(t).

Theorem 1.132 (Second Isomorphism Theorem) - Let G be a group and let H

Theorem 1.133 (Third Isomorphism Theorem) - Let G be a group and let H and K be normal subgroups of G with K

Definition 1.134 - A group G is simple if its only normal subgroups are the trivial subgroups {1} and G.

11. Group Actions

Definition 1.136 - If X is a set and G is a group then a (left)(group) action of G on X is a function *:GxX->X; (g, x)->g*x so that the following conditions and satisfied.

GA1) For all g_1, g_2 in G and x in X, (g_1g_2)*x = g_1 * (g_2 * x)

GA2) For x in X, 1_g * x = x.

Proposition 1.137 - Given an action of G on X, the function a:G->Perm(X); a(g)=a_g is a homomorphism. Conversely, any homomorphism b:G->Perm(X) gives rise to an action of G on X by the rule g*x=b(g)(x) for all g in G and x in X.

Definition 1.140 - Supposed we have an action of a group G on a set X. For x in X,

- the orbit of x is Orb_G(x) = {gx:g in G} which is a subset of X

- the stabiliser of x is Stab_G(x) = {g in G: gx = x} which is a subset of G.

If there is only one orbit then the action is said to be transitive. If the intersection of the stabiliser subgroups is trivial then the action is called faithful.

Theorem 1.141 - for x, y in X

a) Stab_G(x)

b) y in Orb_G(x) if and only if x i