Abstract Algebra - Groups

Binary Operation

'*' on S is a rule assigning each ordered pair (a,b) of elements in S an element a*b that is also in the set S. NOTE: (a,b) ≠ (b,a) BUT {a,b} = {b,a}

How do we know if a binary operation is commutative?

The table will be symmetrical along the diagonal.

What is an induced operation?

(G,*), S⊂G. S is closed under * if for S₁, S₂∈S → S₁*S₂∈S. Then * is an induced operation.

What matrix is a group?

(Mmn(R), +) is a group, where e is the zero matrix of the size m×n (all of the elements in the matrix are zero)

What are two fundamental rules of cosets?

1) All cosets are mutually disjoint. So if g₁H=g₂H, then g₁~g₂ 2) Each coset has |H| many elements. Consequently, |G| = (# of cosets)(|H|)

What are the two important facts about the g.c.d?

1) the g.c.d divides m and the g.c.d. divides n 2) the g.c.d. is the largest among all common divisors of m and n

What are the properties of ~?

1: ∀g∈G, one has g~g 2: ∀g₁,g₂∈G, if g₁~g₂, then g₂~g₁. We know h=g₂⁻¹g, so h⁻¹=g₁⁻¹g₂∈H. 3: ∀g₁,g₂,g∈G, if g₁~g₂, and g₂~g₃, then g₁~g₃.

What does onto mean?

A function f:S→S is said to be onto if to each a∈S, one can find a b∈S such that f(b) = a.

What is a permutation?

A function that is both one-to-one and onto is a permutation of the set S.

Group

A group "G" or (G,*) is a set G with a binary operation * so that * satisfies the following: (1) * is an associative binary operation (2) ∃e∈G so that x*e = e*x = x ∀x∈G (3) For any a∈G, ∃a'∈G so that a'*a = a*a' = e (in other words, a' is the inverse of a, and e is a neutral element)

Abelian Group

A group where * is commutative.

What does a subgroup of a cyclic group look like?

Every subgroup of a cyclic group is itself cyclic.

True or False: All generators of Z₂₀ are prime numbers.

False.

True or false: Every element of every cyclic group generates the group.

False.

True or false: Every group of order ≤4 is cyclic.

False.

True or false: Q under the addition is a cyclic group.

False.

True or false: every abelian group is cyclic.

False.

What is Lagrange's Theorem?

G is a finite group. If g∈G, then o(g)||G|. Let H=<g>=all powers of g. Then |H|=o(g). Suppose G<∞ and H<G. Then |H|||G|.

G=<a>. What type of set is G if o(a)<∞?

G is a finite set. G is of the structure G={e, a, a², a³,...,a^(m-1)} where m=o(a)

G=<a>. What type of set is G if o(a) = ∞?

G is an infinite set. No two exponents h and k can produce the same element in G.

What is a generator?

G is generated by an element g if <g> = G. g is the generator.

What is the General Linear Group?

GLn(R) = {A∈Mn(R) so that ∃B∈Mn(r) with AB=BA=I} where I denotes the n×n identity matrix whose off-diagonal entries are all zero and whose diagnoal entries are all 1.

What is the cancellation law theorem for groups?

Given a group (G,*), the left and right cancellation laws hold. In other words, a*b = a*c → b=c

What is ~?

Given any two elements g₁ and g₂ in G, g₁~g₂ if g₂⁻¹g∈H.

How do we build cells based on ~?

Given g∈G, [g] = gH

Suppose G is a cyclic group generated by a (G=<a>) and o(a)=n. Say b=a^s in G. Then look at H=<b>, the cyclic subgroup generated by the element b. How many elements does H have?

H has n/d elements where d=g.c.d(n,s)

What is the greatest common divisor?

H=<d> for some positive integer d. This value is called the greatest common divisor (g.c.d) of the integers m and n and is denoted by: g.c.d(m,n) = d This value d is also the largest common factor for m and n.

What is the composition of two permutations?

If f and g are both permutations, then their composition f(g(x)) is also a permutation.

What is the order of the element g?

If the order of the cyclic group <g> is finite, then the order of the element g is equal to o(g) or order(g) |<g>| = the number of elements in the finite subgroup <g> o(g) = |<g>| = a finite number m or infinity

Associative

In (S,*), * is associative if (a*b)*c = a*(b*c) for every a,b,c∈S

Commutative

In (S,*), * is commutative if a*b=b*a for every a,b∈S

What is the distribution for the inverse of a*b?

In a group (G,*), (a*b)⁻¹ = b⁻¹ * a⁻¹

How many inverses does an element in a group have?

In a group (G,*), if a∈G is given and assuming that both a' and a'' are inverses of a, then a'=a''. In other words, the inverse of any element a is unique.

What is the solution theorem for groups?

In a group (G,*), the equations a*X = b and Y*c = d have unique solutions X=a'*b and Y=d*c'

How many identity elements does a group have?

In a group (G,*), there is ONLY ONE identity element "e".

What is a subgroup?

Let (G,*) be a group and let H⊂G. H is a subgroup of G if (H,*) is itself a group.

What is a coset?

Let G be a group and H<G. g is an element in G. The (left) coset gH containing the element g is defined as: gH={gh:h∈H} Every coset is a non-empty finite subset of G.

What is a cyclic subgroup?

Let G be a group. Take g∈G. H = <g> is the cyclic subgroup generated by the element g. <g> = {g^n: n=0, ±1, ±2, ±3...} Where g⁰ = e and g^-n = (g⁻¹)^n

How do we know if H is a subgroup of G?

Let H⊂G, where G is a group. Then the subset H is a subgroup if: 1) H is closed under the operation of G. 2) The identity of G must belong to H and serve as its identity. 3) To each a∈H, ∃a⁻¹∈H

What is the sum of h and k modulo n?

Let n be a fixed positive integer and h, k be any integers. Take the remainder of h+k when divided by n. This remainder is the sum of h and k modulo n. Notation: r = h+k (mod n)

Are all matrices groups?

No! Some matrices do not have inverses, such as Mn(R).

Is it possible to find a proper subgroup of Zp if p is a prime number?

No: a proper subgroup is generated by an element that has a g.c.d that is NOT 1 with p. But if p is a prime number, no element in Zp will have a g.c.d greater than 1 with p. Thus, Zp has no proper subgroups if p is a prime number.

Is it possible to have an infinite cyclic group with four generators?

No: all infinite cyclic groups have two generators only.

How do we compute a binary operation given by a table?

Say S = {S₁,S₂,...,Sn). We define * by the table and the rule Si*Sj is the entry located at the ith row and the jth column.

What does one-to-one mean?

Say S is a set. A function f:S→S is said to be one-to-one if f(a) = f(b), so a=b.

What is a symmetric group?

Sn is the set of all permutations σ on the set {1,2,...,n} under the operation of composition. This is a group.

What is a matrix?

The matrix Mmn (R) is the set of all matrices with m rows and n columns. Let A,B∈Mmn(R) such that A=[aij] and B=[bij] (i is the row number, n is the column number) Then A+B = [aij] + [bij]

What is the order of a group?

The order of a group is the number of elements in the group.

What does a finite group's multiplication table look like?

The uppermost two rows are identical, as are the leftmost two rows. Each element appears once and only once per row and per column.

How do we know if a binary operation is associative?

There is no shortcut: we will have to check every combination of elements. As long as we can find one where * is NOT associative, then we can conclude that * is not associative. Otherwise, we can't know until we've tried them all!

Every cyclic group of order >2 has at least two distinct generators.

True.

True or False: If G and G' are subgroups of a group, then their intersection G∩G' must be a subgroup.

True.

True or false: There is at least one abelian group of every finite order >0.

True.

True or false: every cyclic group is Abelian.

True.

What is the Klein-4 Group?

V={e,a,b,c}, forming a commutative table where all the diagonals are e.

How do we multiply matrices?

We define the multiplication AB as follows: (a) AB = [cij] ∈ Mn(R) (b) To compute [cij]: cij = ∑(k=1~n) aik×bkj

On N ( the set of all positive integers), define the operation # by a#b=c, where c is at least 5 more than a+b. Is # a binary operation on N?

Yes. Since a,b∈N and a+b+5≤c, c is always positive and thus c∈N.

What is Zn?

Zn is the set of remainders when dividing by n. [m₁] represents all the integers with remainder m1 [m₂] represents all the integers with remainder m2 [m₁] + [m₂] = [m₁+m₂] = the set of all integers with remainder m₁+m₂

How do we solve [2] + [2] in Z₃?

[2] + [2] = [4] (3k₁ + 2) + (3k₂ + 2) = 3 (k₁+k₂) + 4 = 3 (k₁+k₂+1) +1 So the remainder is 1.

What is Cauchy's Theorem?

if d||G|, then ∃g∈G such that o(g)=d.