Abstract Algebra Exam 2

For every positive integer n, every cyclic group of order n is isomorphic to...

Z_n. Thus, any two cyclic groups of order n are isomorphic to each other.

field

a communitive ring with unity and every nonzero element invertable

Let H be a subgroup of G. H is normal iff aH..

aH = Ha for every a ∈ G.

Every cyclic group of order infinity is isomorphic to , and therefore any two cyclic groups of order infinity...

are isomorphic to eachother

The order of a ^(−l) is the same...

as the order of a

Let H be a subgroup of G. H is normal iff it has the following property: For all a and b in G, ab ∈ H iff...

ba ∈ H.

The order of any element of a finite group...

divides the order of the group.

If f : G → H and H is a homomorphism and is any subgroup of G, then...

f(K) = {f(x) : x ∈ K} is a subgroup of H.

Let f: G → H be a homomorphism with kernel K. Then...

f(a) = f(b) iff Ka = Kb.

et A be a commutative ring with unity or an integral domain respectively. Then A[x] ...

is a commutative ring with unity or an integral domain respectively.

If G = 〈a〉 and b ∈ G, the order of b

is a factor of the order of a

The characteristic of A is the...

least positive integer n such that 1+ 1 + ... + 1 = 0 n times

The center of any group G is a ...

normal subgroup of G.

If G is a group of order n, G is cyclic iff G has an element

of order n

If ord(a) = mk and a^{rk} = e, then...

r is a multiple of m

Division Algorithm for Polynomials

s If a(x) and b(x) are polynomials over a field F, and b(x) ≠ 0, there exist polynomials q(x) and r(x) over F such that a(x) = b(x)q(x) + r(x) and r(x) = 0 or deg r(x) < deg b(x).

Suppose an element a in a group has order n. Then at = e iff...

t is a multiple of n

Quotient Group

Suppose H is a normal subgroup of a group G. The group G/H is called the factor group, or quotient group of G by H.

In an integral domain with nonzero characteristic...

The characteristic is a prime number

The set of all real functions...

Is a ring

kernel of f... Where f is a homomorphism

It is a very important fact that the kernel of f is an ideal of A

Cauchy's Theorem

If G is a finite group, and p is a prime divisor of |G|, then G has an element of order p.

In any integral domain of characteristic p

(a + b)^p = a^p + b^p

Isomorphism of Cyclic Groups

(i) For every positive integer n, every cyclic group of order n is isomorphic to Z_n. Thus, any two cyclic groups of order n are isomorphic to each other. (ii) Every cyclic group of order infinity is isomorphic to Z , and therefore any two cyclic groups of order infinity are isomorphic to each other.

Let f : G → H be a homomorphism (i) The kernel of f is ... (ii) The range of f is...

(i) a normal subgroup of G (ii) A subgroup of H

If G is a group and a ∈G, the following identities hold for all integers m and n"

(i) a^ma^n = a^{m + n} (ii) (a^m)^n = a^{mn} (iii) a^{-n} = a^{(-1)^n} = (a^{n})^{-1}

Let G and H be groups, and f: G → H a homomorphism. Then

(i) f(e) = e, and (ii) f(a−l) = [f(a)]−l for every element a ∈ G.

Ideal

A nonempty subset B of a ring A is called an ideal of A if B is closed with respect to addition and negatives, and B absorbs products in A

Associates

A pair of integers r and s are called associates if they divide each other, that is, if r|s and s|r.

Maximal Ideal

A proper ideal J of a ring A is called a maximal ideal if there exists no proper ideal K of A such that J ⊆ K with J ≠ K (in other words, J is not contained in any strictly larger proper ideal)

Cancellation property

A ring is said to have the ccancellation proberty if ab = ac or ba = ca implies b = c for any elements a,b, and c in the ring if a is not 0. A ring has the cancellation property if it has no divisors of zero

Every cyclic group is

Abelian

Homomrphisms preserve

Abelian, cyclic, finite, each element being its own invers,e having a square root, and bing finitely generated

Prime Ideal

An ideal J of a commutative ring is said to be a prime ideal if for any two elements a and b in the ring, If ab ∈ J then a ∈ J or b ∈ J.

Proper Ideal

An ideal of a ring is called proper if it is not equal to the whole ring.

Integral Domain

An integral domain is a communitive ring with unity having the cancellation property.

Ord(a) = Ord(bab^{-1})

As shown in Chapter 4, Problem H, (bab^{-1})n = ba^nb^{-1}. If ord(a) = p, then (bab^{-1} )^p = ba^pb^{-1} = beb^{-1} = bb^{-1} = e. SInce ord(bab^{-1}) is the least positive integer with this property, ord(bab^{-1}) ≤ p = ord(a). Lemma: If xyx^{-1} = e, then y = e. If or(bab^{-1}) = q, then (bab^{-1})^q = e. Since e = (bab^{-11})^q = ba^qb^{-1}, by the lemma a^q = e. Since ord(a) is the least positive integer with this property, ord(a) ≤ q = ord(bab^{-1}). Since ord(bab^{-1}) ≤ ord(a) and ord(a) ≤ ord(bab^{-1}), ord(a)ord(bab^{-1}).

Subring

B is a subring of A if and only if B is closed with respect to subtraction and multiplication.

Ring

By a ring we mean a set A with operations called addition and multiplication which satisfy the following axioms: (i) A with addition alone is an abelian group. (ii) Multiplication is associative. (iii) Multiplication is distributive over addition.

Every finite integral domain is a ...

Field

Every ideal of F(x) is

Every ideal of F[x] is principal.

Index of H in G

Finally, if G is a group and H is a subgroup of G, the index of H in G is the number of cosets of H in G. We denote it by (G:H). Since the number of elements in G is equal to the number of elements in H, multiplied by the number of cosets of H in G, (G:H) = ord(G)/ord(H).

The family of all the cosets Ha, as a ranges over G, is a partition of G.

First, we must show that any two cosets, say Ha and Hb, are either disjoint or equal. If they are disjoint, we are done. If not, let x ∈ Ha ∩ Hb. Because x ∈ Ha_, x = h_xa for some h_1 ∈ H. Because x ∈ H_b, x = h_2b for some h_2 ∈ H. Thus, h_1a = h_2b, and solving for a, we have a = (h_1^{-1}h_2)b. Thus, a ∈ Hb It follows from Property (1) above that Ha = Hb. Next, we must show that every element c ∈ G is in one of the cosets of H. But this is obvious, because c = ec and e ∈ H; therefore, c = ec ∈ Hc Thus, the family of all the cosets of H is a partition of G. ■

Centrelizer

For any element a ∈ G, the centralizer of a, denoted by Ca, is the set of all the elements in G which commute with a. That is, Ca = {x ∈ G: xa = ax} = {x ∈ G: xax^{−1} = a}

G/H is a homomorphic image of...

G

If G × H is a cyclic group, then...

G and H are both cyclic

The function f: G→G defined by f(x) = x2 is a homomorphism iff G

G is abelian.

H is a normal subgroup of G...

G, then aH = Ha for every a ∈ G.

Principal Ideal

Generated from a single element

G/H with coset multiplication is a...

Group -Associative -Identity -Inverses

Let f: G → H be a homomorphism of G onto H. If K is the kernel of f...

H ≅ G/K

Let H be a normal subgroup of G. If Ha = Hc and Hb = Hd, then...

H(ab) = H(cd).

All nonzero elements in an integral domain ...

Have the same additive order

Endomorphism

Homomorphism from G to G

Direct Products of Rings

If A and B are rings, their direct product is a new ring, denoted by A × B, and defined as follows: A × B consists of all the ordered pairs (x, y) where x is in A and y is in B. Addition in A × B consists of adding corresponding components: (x1, y1) + (x2, y2) = (x1+x2, y1+y2) Multiplication in A × B consists of multiplying corresponding components: (x1, y1) · (x2, y2) = (x1x2, y1y2)

Homomorphisms

If G and H are any groups, and there is a function f which transforms G into H, we say that if is a homomorphic image of G. The function f is called a homomorphism from G to f.

Cyclic Group

If G is a group and a ∈ G, it may happen that every element of G is a power of a. In other words, G may consist of all the powers of a, and nothing else: G = {a^n : n ∈ Z} In that case, G is called a cyclic group, and a is called its generator. We write G = 〈a〉 and say that G is the cyclic group generated by a.

Theorem 5 Let G be a group and H a subgroup of G. Then (i) Ha = Hb iff ab−1 ∈ H and (ii) Ha = H iff a ∈ H

If Ha = Hb, then a ∈ Hb, so a = hb for some h ∈ H. Thus, ab−1 = h ∈ H If ab−1 ∈ H, then ab−1 = h for h ∈ H, and therefore a = hb ∈ Hb. It follows by Property (1) of Chapter 13 that Ha = Hb.

INvertible

If a S is a ring with unity, there may be elementsin A which have a multiplicative inverse. These elements are said to be invertible.

Conjugate

If a ∈ G, a conjugate of a is any element of the form xax^{−1}, where x ∈ G

Division Algorithm

If m and n are integers and n is positive, there exist unique integers q and r such that m = nq + r and 0 ≥ r < n We call q the quotient, and r the remainder, in the division of m by nx.

Suppose an element a in a group has order n. Then at = e iff t is a multiple of n ("t is a multiple of n" means that t = nq for some integer q).

If t = nq, then a^t = a^{nq} = (a^n)^q = e^q = e. Conversely, suppose a^t = e. Divide t by n using the division algorithm: t = nq + r, 0 ≤ r < n. Then e = a^t = a^{nq + r} = ((a^n)^q)(a^r) = e^qa^r = a^r. Thus, a^r = e, where 0 ≤ r <n. If r ≠0, then r is a positive integer less than n, whereas n is the smallest positive integer such that a^n = e. Thus r = 0, and therefore t = nq. ■

No divisors of 0

If the products of two elements in the ring is equal to zero, at least one of the factors is zero

Order of an Element

If there exists a nonzero integer m such that a^m - e, then the order of the element a is defined to be the least positive integer n such that a^n = e. If there does not exist any nonzero integer m such that a^m = e, we say that a has order infinity.

Unity

If there is a neutral element for multiplication, it is called the unity of A, denoted by the symbol 1.

Orbit

If u ∈ A, the orbit of u (with respect to G) is the set O(u) = {g(u): g ∈ G}

Stabalizer

If u ∈ A, the stabilizer of u is the set Gu = {g ∈ G: g(u) = u}, that is, the set of all the permutations in G which leave u fixed.

Divisor of Zero

In any ring, a nonzero element a is called a ivisor of zero if ther is a nonzero element b in the ring such that the product ab or ba = 0

Trivial vs. Nontrivial Ring

Inceidentally, a ring whose only element is 0 is called a trivial ring; a ring with more than one element is nontrivial

Lagrange's Theorem

Let G be a finite group, and H any subgroup of G. The order of G is a multiple of the order of H. In other words, the order of any subgroup of a finite group G is a divisor of the order of G. Let G be a group with a prime number p of elements. If a ∈ G where a ≠ e, then the order of a is some integer m ≠ 1. But then the cyclic group 〈a〉 has m elements. By Lagrange's theorem, m must be a factor of p. But p is a prime number, and therefore m = p. It follows that 〈a〉 has p elements, and is therefore all of G!

If G is a group with a prime number p of elements, then Gi s a cyclic group. Furthermore, any element a ≠ e in G is a generator of G.

Let G be a group with a prime number p of elements. If a ∈ G where a ≠ e, then the order of a is some integer m ≠ 1. But then the cyclic group 〈a〉 has m elements. By Lagrange's theorem, m must be a factor of p. But p is a prime number, and therefore m = p. It follows that 〈a〉 has p elements, and is therefore all of G!

Coset

Let G be a group, and H a subgroup of G. For any element a in G, the symbol aH denotes the set of all products ah, as a remains fixed and h ranges over H. aH is called a left coset of H in G. In similar fashion Ha denotes the set of all products ha, as a remains fixed and h ranges over H. Ha is called a right coset of H in G

Endomorphism

Let G be an abelian group in additive notation. An endomorphism of G is a homomorphism from G to G.

Exponential

Let G be an arbitrary group, with its operation denoted multiplicadvely. Exponential notation is a convenient shorthand: for any positive integer n, we will agree to let a^n = aaaa....a, n times a^{-1} = a^{-1}a^{-1}a^{-1}....a^{-1},, n times and a^0 = e.

Normal Subgroup

Let H be a subgroup of a group G. H is called a normal subgroup of G if it is closed with respect to conjugates, that is, if for any a ∈ H and x ∈ G, xax−l ∈ H.

Parity Group

Let P denote the group consisting of two elements, e and o, with the table We call this group the parity group of even and odd numbers

Kernel

Let f : G → H be a homomorphism. The kernel of f is the set K of all the elements of G which are carried by f onto the neutral element of H. That is, K = {x ∈ G: f(x) = e}

Fundamental Homomorphism Theorem

Let f: G → H be a homomorphism of G onto H. If K is the kernel of f, then H ≅ G/K

Every subgroup of a cyclic group is cyclic. Prove it.

Let m be the smallest positive integer such that a^m ∈ H. We will show that every element of H is a power of a^m, hence a^m is a generator of H. Let a^t be any element of H. Divide t by m using the division algorithm: t = mq + r, 0 ≤ r < m. Then a^t = a^{mq + r} = a^{mq}a^r. Solving for a^r, a^r = (a^{mq})^{-1}a^t = (a^m)^{-q}a^t. But a^m ∈ H and at ∈ H; thus (a^m)^{−q} ∈ H. It follows that a^r ∈ H. But r < m and m is the smallest positive integer such that a^m ∈ H. So r = 0, and therefore t = mq. We conclude that every element a^t ∈ H is of the form a^t = (a^m)^q, that is, a power of a^m. Thus, H is the cyclic group generated by a^m. ■

Theorem 3: Powers of a, if a has finite order if the order of a is n, there are exactly n different powers of a, namely, a^0, a, a^2, a^3, ..., a^{n−1}. What this theorem asserts is that every positive or negative power of a is equal to one of the above, and the above are all different from one another. Before going on, remember that the order of a in n, hence a^n = e and n is the smallest positive integer which satisfies this equation.

Let us begin by proving that every power of a is equal to one of the powers a^0, a, a^2 a^3, ..., a^{n−1} . Let am be any power of a. Use the division algorithm to divide m by n: m = nq + r 0 ≤ r < n. Then a^m = a^{nq + r} = (a^{nq}a^{r}) = (a^{n})^{q}a^r = e^qa^r = a^r Thus, a^m = a^r, and r is one of the integers 0,1, 2, ..., n − 1. Next, we will prove that a^0, a, a^2 a^3, ..., a^{n−1} are all different, Suppose not; suppose a^r = a^s, where r and s are distinct integers between 0 and n - 1. Either r < s or s < r, say s < r. Thus, 0 ≤ s < r < n, and consequently, 0 < r − s < n (1) Then a^r(a^s)^{-1} = a^s(a^s)^{-1}, hence a^{r - s} = e. But a has order infinity, and this means that am is not equal to e for any integer m expect 0. Thus, r − s = 0. so r = s. ■

Prove: If a ∈ Hb, then Ha = Hb

Let x ∈ Ha; this means that x = h_2a for some h_2 ∈ H. But a = h_1b, so x = h_2a = (h_2h_1)b, and the latter is clearly in H_b. This proves that every x ∈ H_a is in H_b; analogously, we may show that every y∈ H_b is in H_a, and therefore H_a = H_b.

Commutative Ring

Multiplication is Commutative

Every subgroup of an abelian group...

Normal

Commutor

Our next example may bring out this idea even more clearly. Let G be an arbitrary group; by a commutator of G we mean any element of the form aba−1b−1 where a and b are in G. The reason such a product is called a commutator is that aba−1b−1 = e iff ab = ba

Clam. Let a and b be any elements of a ring A. (i) a0 = 0 and 0a = 0 (ii) a(−b) = − (ab) and (−a)b = −(ab) (iii) (−a)(−b) = ab

Proof. aa + 0 = aa = a(a + 0) = aa + a0 -> 0 = a0 Thus, aa + 0 = aa + a0. By Condition (1) above we may eliminate the term aa on both sides of this equation, and therefore 0 = a0. To prove (ii), we have a(-b) + ab = a[(-b) + b] by the distributive law = a0 = by part(i) Thus, a(−b) + ab = 0. By Condition (2) above we deduce that a(−b) = − (ab). The twin formula (−a)b = − (−ab) is deduced analogously. We prove part (iii) by using part (ii) twice: (−a)(−b) = −[a(−b)] = −[ − (ab)] = ab ■

Prove: If Ha is any coset of H, there is a one-to-one correspondence from H to Ha.

The most obvious function from H to Ha is the one which, for each h ∈ H, matches h with ha. Thus, let f: H → Ha be defined by f(h) = ha Remember that a remains fixed whereas h varies, and check that f is injective and surjective. f is injective: Indeed, if f(h_1) = f(h_2), then h_1a = h_2a, and therefore h_1 = h_2. f is surjective, because every element of Ha is of the form ha for some h ∈ H, and ha = f(h). Thus, f is a one-to-one correspondence from H to Ha, as claimed. ■

Order of a Group

The order of a group G is the number of element in G, denoted by |G|.

Parity Ring

The parity ring P consists of two elements, e and o, with addition and multiplication given by the tables We should think of e as "even" and o as "odd," and the tables as describing the rules for adding and multiplying odd and even integers. For example, even + odd = odd, even times odd = even, and so on.

Ring of the Integers

The set Z with conventional addition and multiplication

Relatively Prime

Two integers m and n are said to be relatively prime if they have no prime factors in common.

Let a be an element of order m in any group G. For what values of k is ord(a^k) = m?

When k and m are relatively prime.

If the order of a is not a multiple of ra, then...

the order of a^k is not a multiple of m

The order of ab is the same as...

the order of ba

It is an important fact that if A is a commutative ring with unity...

then J is a maximal ideal of A if A/J is a field

1 If a is a power of b, say a = bk, then..

〈a〉 ⊆ 〈b〉.