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〉.