Cardinality and Algebraic Structures

Cardinality and Algebraic Structures Dr Tijl De Bie Dept. Eng. Maths. email: tijl.debie@gmail.com

Contents Part I (weeks 1-7) • 1 Introduction • 2 Combinatorics, permutations and combinations. • 3 Algebraic Structures and matrices: Homomorphism, isomorphism, group, semigroup, monoid, rings, fields • 4 Lattices and Boolean algebras • If time remains: some illustrations of the use of group theory in cryptography Part II (weeks 8-12) • Vector spaces

Introduction • Computer programs frequently handle real world data. • This data might be financial e.g. processing the accounts of a company. • It may be engineering data e.g. from sensors or actuators in a robotic system. • It may be scientific data e.g. weather data or geological data concerning rock strata. • In all these cases data typically consists of a set of discrete elements. • Furthermore there may exist orderings or relationships among elements or objects. • It may be meaningful to combine objects in some way using operators. • We hope to clarify our concepts of orderings and relationships among elements or objects • We look at the idea of formal structures such as groups, rings and and formal systems such as lattices and Boolean algebras

Number Systems • The set of natural numbers is the infinite set of the positive integers. It is denoted N and can have different representations: {1,2,3,4,........} {1,10,11,100,101,.....} are alternative representations of the same set expressed in different bases. Nm is the set of the first m positive numbers i.e. {1,2,3,4, ......,m}. N0 is the set of natural numbers including 0 i.e. {0,1,2,3,5,....} • Q denotes the set of rational numbers i.e. signed integers and fractions {0,1,-1,2,-2,3,-3,....,1/2,-1/2,3/2,-3/2,5/2, -5/2,....,1/3,-1/3,2/3,-2/3,........} • R is the set of real numbers i.e. the coordinates of all the points on a line. • Z is the set of all integers, both positive and negative {0,1,-1,2,-2,3,-3,......}

2 Combinatorics: Permutations • A permutation of the elements of a set A is a bijection from A onto itself. • If A is finite we can calculate the number of different permutations. Suppose A={a1,...,an} n choices n-1 choices 1 choice a1 a2 an total number of ways of filling the n boxes n x (n-1)x(n-2)x(n-3)..............x1=n! nPn=n! eg a possible permutation of {1,2,3,4,5,6} is

Composition of Permutations • If :A A and :A A are permutations of A then the composition or product .of and satisfies for all x in A .x)= (x)) Notice that since both and are bijections from A into A so is .In other words . is a permutation of A. • Example: Let A={1,2,3,4,5,6} then two possible permutations are For .we have that

Cyclic Permutations A cyclic permutation on a set A of n elements has the form where : For shorthand we often write  is said to be a k cycle Example or (6 1 4) is a cyclic permutation Two cyclic permutations and are said to be disjoint if e.g. (4 5 2) and (3 1 6) are disjoint

Notice that Other examples are or Can you spot a product of disjoint cyclic permutations equivalent to the following permutation ?

Theorem: Every permutation of a finite set A can be expressed as a combination of disjoint cycles. Structure underlying permutations Note that the following hold: (1) The product of two permutations is a uniquely determined permutation of the same set. (2) The composition of permutations is associative. (3) The permutation is called the identity permutation andhas the property that (4) For every permutation there is an inverse such that

Combinations • When we think about combinations we do not allow repeats and unlike permutations we do not consider order. • Combinations look at the number of different ways of picking a subset of k elements from a set of n elements. • Think of the number of ways of picking a list of k distinct elements of n no. of choices n n-1 n-k-2 n-k-1 places = n(n-1)(n-2) ........... (n-k-1) = n!/(n-k)! For each possible list there are k! permutations so since we are not interested in order we should divide the above by k!. C(n,k) = Cnk = n!/(n-k)!k!

Example: Choosing 2 elements from {a,b,c,d} {a,b},{a,c},{a,d}, {b,c},{b d},{c,d} C(4,2)= 4!/(2! 2!) =6 Combinations with Repetitions We could also consider combinations with repetitions. With repetitions the number of distinct combinations of k elements chosen from n is: C(n+k-1,k)= (n+k-1)!/k!(n-1)! Number of different throws of 2 identical dice (1 1)(2 2)(3 3)(4 4)(5 5)(6 6) (1 2)(1 3)(1 4)(1 5)(1 6) (2 3)(2 4)(2 5)(2 6) (3 4)(3 5)(3 6)(4 5)(4 6)(5 6) C(7,2)=21

Algebraic Structures • When we consider the behaviour of permutations under the composition operation we noticed certain underlying structures. • Permutations are closed under this operation, they exhibit associativity, an identity element exists and an inverse exists for each permutation • These properties define a general type of algebraic structure called a group. • In this section we shall look at groups in more detail as well as other similar algebraic structures such as semigroups and monoids. • Later we will progress to consider more complex algebraic structures such as rings, integral domains and fields. • We will see that many real life situations are examples of these algebraic structures

A group or is a set G with binary Groups operation which satisfies the following properties 1. is a closed operation i.e. if and then 2. this is the associative law 3. G has an element e, called the identity, such that 4. there corresponds an element such that Example: The set of all permutations of a set A onto itself is group (called the symmetric group Sn for n elements).

Group of Symmetries of a Triangle l Consider the triangle X O Y Z n m We can perform the following transformations on the triangle 1=identity mapping from the plane to itself p=rotation anticlockwise about O through 120 degrees q=rotation clockwise about O through 120 degrees a=reflection in l b=reflection in m c=reflection in n

Let denote transformation y followed by transformation x for x and y in {1,p,q,a,b,c} So for example l l l Y X X a p O O O X Z m n m Y Z n m Z Y n Notice the table is not symmetric

Other examples of a group • The set of all permutations onto itself is a group (called the symmetric group Sn) • The sets of all invertible nxn matrices forms a group under ordinary matrix multiplication (called GL(n), the general linear group) • The quaternion group: Let G={I,-I,J,-J,K,-K,L,-L} where • I=[ ], J=[ ], K=[ ] , L=[ ] 1 00 1 j 00 -j 0 1-1 0 0 jj 0

Order of a group • A finite group is a group where G is finite • The order of a finite group is |G| • For example if G is the set of permutations of a set A with n elements then the order of G is n!

Abelian Groups If is a group and is also commutative then is referred to as an Abelian group (the name is taken from the 19’th century mathematician N.H. Abel) is commutative means that Examples: and are abelian groups. Why is not a group at all?

Modular arithmetic • Recall a=b mod p iff p|a-b • Notice a=b mod p iff a=kp+b for some integer k •  a=b mod p implies p|a-b impliesa-b=kp implies a=kp+b •  a=kp+b implies a-b=kp impliesp|a-b implies a=b mod p

Modular addition • Modular addition mod 6:

Modular multiplication • Modular multiplication mod 7:

Modular multiplication • Modular multiplication mod 6:

Modular multiplication • Not a group! (Why not?) • Which subset of {1,2,3,4,5} does form a group?

Modular multiplication Theorem: If n>=2 and n|p then n has no inverse under multiplication mod p Prove it! The subset of {1,…,p-1} relatively prime to p is a group under multiplication mod p denoted Zp* We will clarify this on the next slides…

Modular arithmetic • Recall Euclid’s algorithm to find the gcd of x and y: x=k1y+r1 y=k2r1+r2 r1=k3r2+r3 … rn-2 =kn-1rn-1+rn rn-1=knrn From this… Theorem: There exist integer a and b such ax+by=gcd(x,y) The old remainder is divided by the new one repeatedly until the remainder is 0 The gcd is the last non zero remainder

Modular arithmetic • An element n has an inverse n-1 under multiplication mod p for which n. n-1 =1 mod p if and only if (iff) n is relatively prime to p. • Prove this! • Clearly then if p is prime then every element will have an inverse.

Groups in logic Consider exclusive or defined by A⊕ B ≡(¬A∧ B)∨ (A∧ ¬B) • {t,f} is an abelian group under exclusive or. • What is the identity? • What is the inverse of t (and f)?

Two show that an algebraic system is a group we must show that it satisfies all the axioms of a group. Question: Let be a Boolean algebra so that A is a set of propositional elements, is like ‘or’, is like ‘and’ and is like ‘not’. Show that is an abelian group where Answer: (1) Associative since prove this ? (2) Has an identity element 0 (false) since (3) Each element is its own inverse (4) The operation commutes prove this ?

Iterated operations • a=a1 • a◦a=a1 • a◦a◦a=a2 • a◦a…◦a=ak • (Why is this unambiguously defined?)

Cyclic groups A group G is cyclic if there exists a∈G such that for any b∈G there is an integer k≥0 such that ak=b. I.e. Every element of G is some power of a. Element a is called the generator of G denoted G=<a> Example: • <{1,-1},×>=<-1> since –12=1, -13=-1

Order of a cyclic permutation group • (1 2 … p) • Show that the order is equal to p • [Show by making a drawing…]

Weaker structures • An Abelian group is a strengthening of the notion of group (i.e. requires more axioms to be satisfied) • We might also look at those algebraic structures corresponding to a weakening of the group axioms Semigroup ⊆ monoid ⊆ group ⊆ Abelian Group

Semigroup is a semigroup if the following conditions are satisfied: 1. is a closed operation i.e. if and then 2. is associative • Example: The set of positive even integers • {2,4,6,.....} under the operation of ordinary addition • since • The sum of two even numbers is an even number • + is associative The reals or integers are not semigroups under - why?

is a monoid if the following conditions Monoid are satisfied: 1. is a closed operation i.e. if and then 2. is associative 3. There is an identity element Examples: Let A be a finite set of heights. Let be a binary operation such that is equal to the taller of a and b. Then is a monoid where the identity is the shortest person in A is a monoid: is associative, true is the identity, but false has no inverse is a monoid: is associative false is the identity, but true has no inverse

Properties of Algebraic Structures properties Theorem: (unique identity) Suppose that is a monoid then the identity element is unique Proof: Suppose there exist two identity elements e and f. [We shall prove that e=f] Theorem: (unique inverse) Suppose that is a monoid and the element x in A has an inverse. Then this inverse is unique. Proof: ??

Properties of Groups Theorem (The cancellation laws): Let be a group then (i) (ii) Proof: (i) Suppose that then by axiom 3 a has an identity and we have that (ii) is proved similarly Theorem (The division laws): Let be a group then (i) (ii) Proof ??

Theorem (double inverse) :If x is an element of the group then Proof: Theorem (reversal rule) If x and y are elements of the group then Proof ??

For an arbitrary element of a group we can define functions and such that Theorem: and are permutations of G Proof: Consider [prove 1-1] suppose for x,y in G [Prove onto] For any y in G Corollary: In every row or column of the multiplication table of G each element of G appears exactly once.

is a subgroup of the group if Subgroups and is also a group Examples: is a subgroup of is a subgroup of Test for a subgroup Let H be a subset of G. Then is a subgroup of iff the following conditions all hold: (1) (2) H is closed under multiplication (3) For every group , and are subgroups is called the trivial subgroup of a proper subgroup of is a subgroup different from G A non-trivial proper subgroup is a subgroup equal neither to or to

Cosets . Then the left coset of H with respect to a is the set of elements: Consider a set A with a subset H. Let This is denoted by Similarly the right coset of H with respect to a is and is denoted by Example: Let A be the set of rotations and . Let then {0º,120º,240º} which is the right coset with respect to

Normal Subgroups Let be a subgroup of . Then is a normal subgroup if, for any , the left coset is equal to the right coset is a normal subgroup where e.g. Theorem: In an Abelian group, every subgroup is a normal subgroup

Coset cardinality Theorem: For any H subset of G and any a in G |a•H|=|H| Proof: By definition of Coset |a•H|≤|H| Now suppose |a•H|<|H| then there must exist b and c distinct elements of H such that a•b=a•c. But by the cancellation law this implies that b=c which is a contradiction. Hence |a•H|=|H|

Coset partitioning Theorem: Let a,b∈G and let H be a subgroup of G then either:a•H=b•H or: a•H∩ b•H=∅ Proof: Suppose a•H∩ b•H≠∅ then there exist s and t in H such that a•s=b•t. In this case a= b•t•s-1 and for an arbitrary x in Ha•x= b•t•s-1•x Now by the inverse axiom and closure,t•s-1•x∈H and hence b•t•s-1•x∈b•H, therefore a•x∈b•H so that a•H⊆b•H Similarly we can show that b•H⊆a•H Hence if the two cosets are not disjoint then b•H=a•H

LeGrange’s theorem Theorem: Let H be a subgroup of finite group G, then the cardinality of H evenly divides the cardinality of G (i.e |H| | |G|) Proof Let |G|. Now for each element ai of G we can generate a coset ai•H. Now notice that ai∈ai•H because since H is a subgroup, e∈H and ai•e= ai Suppose there are m distinct cosets of H then picking one representative ai from each this means that:G= a1•H∪ a2•H ∪ a3•H … ∪ am•H

LeGrange’s theorem Now by the previous theorem it follows that since these m cosets are distinct then they must be disjoint. Hence, |G|=|a1•H|+ |a2•H| + |a3•H| … + |am•H| Also by the cardinality theorem for cosets they all have the same cardinality, namely |H|. Hence, |G|=m.|H| as required

Order of an element • Let i be the smallest integer such that ai=e where a is an element of group G and e is the identity element. • If i exists we call it the order of a. • Otherwise we say that a has infinite order.

Subgroup generated by an element Theorem: For any element a of G with finite order the set:H={aj: for some integer j}is a subgroup of G. Notice: if i is the order of element a then • ai=e • ai+1=e•a=a1 • ai+2= a •a =a2 • ai+n=an

Example • Let σ=(1 2 3 4), a permutation of 4 elements • Then {σ, σ2, σ3, σ4} is a subgroup of the group of permutations of {1,2,3,4} • The order of σ is 4 • [Work it out!]

Order of elements in finite groups • If the group G is finite then all elements of G have finite order: • For any a∈G, since G is finite there must exist i<j such that ai=aj • a•ai-1=a•aj-1 cancellation law implies ai-1=aj-1 • Repeated application of the cancellation law gives a=aj-i+1 • a•e=a•aj-i implies e=aj-i

Corollary of LeGrange Theorem: The order of every element of a finite group G, divides the order of G Proof... Every element of G has finite order n and hence generates a subgroup of order n. Hence by LeGrange’s theorem n divides |G|

Cardinality and Algebraic Structures